MODELING OF PERIODIC DIELECTRIC STRUCTURES

(ELECTROMAGNETIC CRYSTALS)

by

John David Shumpert

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy(Electrical Engineering)

in The University of Michigan2001

Doctoral Committee:Professor Linda P. B. Katehi, ChairAssociate Professor Kamal SarabandiEmeritus Professor Thomas B. A. SeniorProfessor Wayne E. Stark

“Almighty God, Who hast created man in Thine own Image, and made him a

living soul that he might seek after Thee, and have dominion over Thy crea-

tures, teach us to study the works of Thy hands, that we may subdue the earth

to our use, and strengthen our reason for Thy service; and so to receive Thy

blessed word, that we may believe on Him whom Thou hast sent, to give us the

knowledge of salvation and the remission of our sins. All which we ask in the

name of the same Jesus Christ, our Lord.”

— James Clerk Maxwell (1831–1879)

c© John David Shumpert 2001All Rights Reserved

Sola Deo Gloria

ii

ACKNOWLEDGMENTS

I am grateful to Professor Linda P. B. Katehi for serving as my dissertation chair and for

her continued support of my work. I also appreciate Professors Kamal Sarabandi, Thomas

B. A. Senior, and Wayne E. Stark for their welcomed comments and suggestions and for

graciously giving of their time to serve on my dissertation committee.

I am grateful to the National Science Foundation without whose financial support, in

the form of the National Science Foundation Graduate Research Fellowship, some of my

graduate work would not have been possible. I also gratefully acknowledge the support of

the Department of Defense Research and Engineering (DDR&E) Multidisciplinary Univer-

sity Research Initiative (MURI) on “Low Energy Electronics Design for Mobile Platforms”

managed by the Army Research Office (ARO) under Grant DAAH04–96–1–0377.

The completion of an extended period of directed research that is required to obtain

an advanced degree is the product of hard work, perseverance, and the continued support

of family, friends, and my fellow comrades-in-arms. The Radiation Laboratory at the Uni-

versity of Michigan is not simply the group of offices, laboratories, and classrooms that

comprise the major portion of the third floor of the EECS Building; it is the collection

of faculty, staff, and students who live and work together every day. In particular, staff

members Mike Angelini, Michelle Shepherd, Carol Truskowski, and Patti Wolfe were vaults

of hidden knowledge for myself and the other graduate students in the Lab. Carol, your

indefatigable spirit is an invaluable asset to the graduate students in the Lab. I would like

to thank my comrades-in-arms, current and former University of Michigan Radiation Lab

graduate students Lars Andersen, Scott Barker, Andy Brown, Mark Casciato, Bill Chappell,

Rashaunda Henderson, Katherine Herrick, Stephane Legault, Sergio Pacheco, Ron Reano,

and Kyoung Yang, for the hours of enjoyable conversation and fun. I am thankful to Bill

Chappell, Saqib Jalil, and Tom Ellis, for their help with the fabrication and design aspects

of the slot antenna work.

iii

I would like to especially thank Mark Casciato, Katherine Herrick, Stephane Legault,

and Sergio Pacheco, without whom my short time at the University of Michigan would

have seemed an eternity. I will dearly miss our many “not-so-technical” discussions. Your

friendship has been and remains an invaluable asset to me and to my family.

I am very blessed to have the love and support of my family. Dad, I am now your

colleague; you have always been my biggest fan. Mom, I thank you for your ever encouraging

and joyful spirit. Simple words seem inadequate to express my love for you both and my

appreciation for your continual prayers throughout this journey. Jim and Stephanie, your

confidence in my ability to persevere was a welcomed voice at the other end of the telephone.

Caleb, I thank you for your unconditional love for Daddy.

Breanna, to you I express my deepest appreciation. Your prayers, unconditional

love, constant encouragement, and unwavering devotion sustained me during the

many long days of deriving pages of equations or sitting alone in front of a com-

puter screen desperately debugging long Fortran codes. May our Lord, Jehovah,

the God of the Bible, bless you with the desires of your heart for providing me

with the strength and perseverance to complete this work.

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PREFACE

Electromagnetic band-gap (EBG) materials, often termed electromagnetic crystals or

photonic band-gap (PBG) materials, are found to have unique properties that are advanta-

geous for applications involving semiconductor integrated circuits (ICs). Preliminary results

suggest that at microwave and millimeter-wave frequencies the propagation characteristics

of these materials can be manipulated by carefully designing and fabricating periodic struc-

tures composed of regions of differing dielectric constants. Researchers from the diverse

fields of classical electromagnetics, solid-state physics, optics, material science, condensed

matter physics, and semiconductor physics, are actively contributing to this rediscovered

field of physics. Active areas of electromagnetic and photonic crystal research include but are

not limited to microwave and millimeter-wave antenna structures, quasi-optical microwave

arrays, photonic crystal integrated circuits, high- and low-Q electromagnetic resonators,

quantum optical electromagnetic cavity effects, and optical nano-cavities. In addition, sonic

band-gap materials or artificial acoustic crystal substrates, are being developed and could

impact sonar.

Recent attention in the field of EBGs has focused on the elimination of surface-wave

formation in planar microwave and millimeter-wave antenna applications. A number of

numerical techniques, including finite difference methods, finite element methods, spectral

domain analysis, and integral equation (IE)/moment methods (MoM), have been imple-

mented to help understand the properties of surface and leaky waves on a layered, periodic

structure. The IE/MoM approach is a useful approach because information about the

physical system (propagation constants, mode structure, etc.) can be obtained easily and

directly from the formulation. A full-wave IE/MoM code has been developed to determine

the band structure (propagating modes) of periodic one- and two-dimensional inhomoge-

neous dielectric regions for specific use in planar antenna applications. This work derives

the integral equation approach along with several examples showing the unique spectral

characteristics of EBGs.

v

TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTER

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 One-Dimensional Periodic Dielectric Structures . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Propagation of Waves in a Periodic Medium . . . . . . . . . 13

2.3 Plane Wave Expansion Method . . . . . . . . . . . . . . . . . . . . 142.3.1 Analytical Techniques . . . . . . . . . . . . . . . . . . . . . 142.3.2 Matrix Solution of the Eigensystem . . . . . . . . . . . . . . 18

2.4 Formulation of Integral Equations . . . . . . . . . . . . . . . . . . . 232.4.1 Derivation of Electric Field Integral Equation . . . . . . . . 232.4.2 Equivalent Polarization Currents . . . . . . . . . . . . . . . 242.4.3 Method of Moments Formulation . . . . . . . . . . . . . . . 282.4.4 Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . 392.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Two-Dimensional Periodic Dielectric Structures . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Plane Wave Expansion Method . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Analytical Techniques . . . . . . . . . . . . . . . . . . . . . 473.2.2 Matrix Solution of the Eigensystem . . . . . . . . . . . . . . 53

3.3 Formulation of Integral Equations . . . . . . . . . . . . . . . . . . . 56

vi

3.3.1 Derivation of Electric Field Integral Equation . . . . . . . . 563.3.2 Method of Moments Formulation . . . . . . . . . . . . . . . 613.3.3 Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Effective Medium Theory (EMT) . . . . . . . . . . . . . . . . . . . 673.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Scattering From an Inhomogeneous Doubly Periodic DielectricLayer Above a Layered Medium . . . . . . . . . . . . . . . . . . . . . 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Formulation of Integral Equations . . . . . . . . . . . . . . . . . . . 74

4.2.1 Derivation of Electric Field Integral Equation . . . . . . . . 744.2.2 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 824.3.1 Impedance Matrices . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Excitation Matrices . . . . . . . . . . . . . . . . . . . . . . . 91

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.5 Series Acceleration Techniques for the Periodic, Free-Space Green’s

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.5.1 Acceleration Techniques Applied in This Work . . . . . . . 954.5.2 Acceleration Techniques Used Elsewhere . . . . . . . . . . . 107

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Radiation Properties of a Rectangular Microstrip Patch on a Uni-axial Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Equivalent Uniaxial Modeling . . . . . . . . . . . . . . . . . . . . . 113

5.2.1 Hashin-Shtrikman Model . . . . . . . . . . . . . . . . . . . . 1145.2.2 Licktenecker Model . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Plane Wave Reflection Coefficients for a Grounded Uniaxial Layer . 1195.4 Rectangular Patch Antenna on a Positive Uniaxial Substrate . . . . 1245.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6 Recommendations For Future Work . . . . . . . . . . . . . . . . . . . 1306.1 Arbitrarily Shaped Dielectric Function . . . . . . . . . . . . . . . . 1306.2 Extensions of Effective Medium Theory . . . . . . . . . . . . . . . . 1326.3 Noncommensurate Periodicities . . . . . . . . . . . . . . . . . . . . 1336.4 Extensive Parametric Study of Antenna Performance Near a Periodic

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

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LIST OF TABLES

Table2.1 Normalized TEx propagation constant kx0a for increasing number of Floquet

modes for f0=fa/c=1, εr=8.9, b=0.3545a, and Nx=40 . . . . . . . . . . . . 352.2 Normalized TEx propagation constant kx0a for increasing number of un-

knowns for f0=fa/c=1, εr=8.9, b=0.3545a, and Np=51 . . . . . . . . . . . 362.3 Normalized TMx propagation constant kx0a for increasing number of Floquet

modes for f0=fa/c=1, εr=8.9, b=0.3545a, and Nx=40 . . . . . . . . . . . . 372.4 Normalized TMx propagation constant kx0a for increasing number of un-

knowns for f0=fa/c=1, εr=8.9, b=0.3545a, and Np=51 . . . . . . . . . . . 384.1 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relative

permittivity εr=2.56 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for Nx=Ny=Nz=4, Np=Nq=61, φ0=45, θ0=45, and a=0.25λ0 93

4.2 Summary of series transformations . . . . . . . . . . . . . . . . . . . . . . . 954.3 Convergence of higher order Shanks’ transforms of Leibnitz series as a func-

tion of order and number, π=3.14159265358979 . . . . . . . . . . . . . . . . 984.4 Error for various (N, k) Shanks’ transforms for Leibnitz series . . . . . . . . 995.1 Minimum achievable anisotropic ratio (AR) . . . . . . . . . . . . . . . . . . 1185.2 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relative

permittivity εr=2.56 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for 192 unknowns and 288 unknowns for Np=Nq=61, φ0=45,θ0=45, and a=0.25λ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for 188 unknowns, Np=Nq=61, φ0=60, θ0=30, and a=0.25λ0 123

5.4 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=6.15 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for 192 unknowns and 288 unknowns for Np=Nq=61, φ0=45,θ0=45, and a=0.25λ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.5 Dependence of resonant frequency on substrate permittivity . . . . . . . . . 126

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LIST OF FIGURES

Figure2.1 One dimensional lattice of dielectric slabs of width b in a periodic lattice

with period a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Full band structure for a normally incident mode in a periodic array of di-

electric slabs with b/a=0.1 and εr=8.9 . . . . . . . . . . . . . . . . . . . . . 192.3 Band structure for one dimensional lattice of dielectric slabs of with filling

fraction b/a=0.5 for (a) εb=εa=13, (b) εb=12, εa=13, and (c) εb=1, εa=13 202.4 Band structure (TMx) for lowest order mode of a periodic array of dielectric

slabs as a function of filling fraction b/a for normal incidence and εr=8.9 . 212.5 Band structure (TMx) for lowest order mode of a periodic array of dielectric

slabs as a function of dielectric constant εr for normal incidence and fillingfraction b/a=0.3545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Band structure (TMx) for lowest order mode of a periodic array of dielectricslabs as a function of off-axis propagation constant ky0 for filling fractionb/a=0.3545 and εr=8.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Representative dielectric slab of width b with period a . . . . . . . . . . . . 252.8 Piecewise constant and piecewise linear basis functions used in this work . . 292.9 Band structure (TEx, TMx) of a periodic array of dielectric slabs for b/a=

0.3545, εr=8.9, and off-axis propagation constant ky0=1/a computed usingmethod of moments technique (MoM) with Np=51 and plane wave expansion(PWE) solution with Np=63 . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.10 Microstrip mounted on 1-D periodic dielectric substrate with period a, fillingfraction b/a=0.5, and height t . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.11 Measured frequency response of a microstrip mounted on a 1-D grating and2-D lattice substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.12 Simulated frequency response of HFSS and Libra simulations of a microstripmounted on a 1-D grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.13 Microstrip mounted on (a) 1-D grating, (b) finite 1-D grating, (c) 2-D latticeof square holes, and (d) 2-D lattice of circular holes . . . . . . . . . . . . . . 42

2.14 Simulated frequency response of a microstrip mounted on a 1-D grating, finite1-D grating, 2-D lattice of square holes, and 2-D lattice of circular holes . . 43

2.15 Microstrip mounted on two-dimensional periodic dielectric substrate withperiod a, cell-to-cell distance c=a/

√2, hole diameter d, and height t . . . . 43

3.1 Cross-sectional view of two-dimensional array of dielectric rods of diameterb in a periodic square lattice with period a . . . . . . . . . . . . . . . . . . 47

ix

3.2 Unit cell dimensions for (a) rectangular dielectric rods and (b) circular di-electric rods where for the rectangular rod, a and c are the unit cell sizes, band d are the element edge lengths, and for the circular rod, a is the unitcell size and b is the diameter . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Unit cell dimensions for (a) rectangular dielectric air columns and (b) circularair columns where for the rectangular column, a and c are the unit cell sizes,b and d are the element edge lengths, and for the circular column, a is theunit cell size and b is the diameter . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Band structure (TMz) for a normally incident mode in a periodic array ofdielectric rods with b/a=0.3545 and εr=8.9 . . . . . . . . . . . . . . . . . . 54

3.5 Band structure (TEz) for a normally incident mode in a periodic array of aircolumns with b/a=0.6455 and εr=8.9 . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Equivalent structures for Figure 3.2(a) and Figure 3.3(a) . . . . . . . . . . . 553.7 Normalized TMz propagation constant kx0a as a function of d/c for f0=

fa/c=0.5, εr=10.2, ky=0, and b/a=0.2 . . . . . . . . . . . . . . . . . . . . . 563.8 Representative dielectric rod of diameter b in a periodic square lattice with

period a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.9 Band structure (TMz, TEz) of a periodic lattice of dielectric columns with

filling fraction b/a=d/c=0.3545, εr=8.9, Nx=Ny=15, and Np=Nq=31 . . . 663.10 Effective medium theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.11 Exact band structure and effective medium theory approximation for periodic

lattice of dielectric columns with filling fraction b/a=d/c=0.3545, dielectricconstant εr=8.9, and period-to-wavelength ratio α = a/λ0=0.5 for (a) TMz

case and (b) TEz case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.12 Exact band structure and effective medium theory approximation for peri-

odic lattice of dielectric columns with filling fraction b/a=d/c=0.2, dielectricconstant εr=8.9, and period-to-wavelength ratio α = a/λ0=0.5 for (a) TMz

case and (b) TEz case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1 Doubly periodic dielectric layer with period a in the x direction and period

c in the y direction over a layered medium . . . . . . . . . . . . . . . . . . . 754.2 Coordinate system (kDB) for incident and scattered field . . . . . . . . . . 764.3 Doubly periodic dielectric layered unit cell . . . . . . . . . . . . . . . . . . . 784.4 Plane wave reflection from layered medium . . . . . . . . . . . . . . . . . . 814.5 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relative

permittivity εr=2.56 and dielectric thickness t=0.15λd as a function of θ0 forNx=Ny=Nz=3, Np=Nq=61, and a=0.1λ0 . . . . . . . . . . . . . . . . . . . 92

4.6 Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of θ0 forNx=Ny=Nz=3, Np=Nq=61, and a=0.1λ0 calculated using the multilayeredsolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.7 Relative error as a function of floating point operations for computing variousorder Shanks’ transforms and the direct sum of the inner sum of (4.80) . . 101

4.8 Relative error as a function of floating point operations for computing variousorder Shanks’ transforms, Kummer’s transforms, and the direct sum of theinner sum of (4.80) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

x

5.1 (a) Periodic dielectric layer over a layered medium and (b) equivalent uniax-ial medium with transverse permittivity εt and axial permittivity εz over alayered medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Effective transverse permittivity as a function of filling fraction for εb=10εa 1155.3 Effective transverse permittivity as a function of filling fraction for εa=10εb 1165.4 Uniaxial nature of sapphire modeled by (a) periodic dielectric rods with

ε2=12.1ε1 and b/a=0.95 and (b) periodic dielectric veins with ε2=16.0ε1and b/a=0.30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5 Geometry for a rectangular patch on a grounded dielectric substrate . . . . 1255.6 Directivity patterns for reference patch with AR=1 and AR=0.5 . . . . . . 1286.1 Optimized frequency selective layer (volume) . . . . . . . . . . . . . . . . . 1316.2 Dielectric and/or metallic loading . . . . . . . . . . . . . . . . . . . . . . . . 1316.3 Frequency selective supercell . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.4 Higher-order effective medium theory . . . . . . . . . . . . . . . . . . . . . . 1336.5 Noncommensurate periodicities . . . . . . . . . . . . . . . . . . . . . . . . . 133A.1 Loaded and equivalent transmission line circuits . . . . . . . . . . . . . . . . 138B.1 Direct lattice defined using the primitive lattice vectors a and b and the

reciprocal lattice defined using the primitive reciprocal lattice vectors a∗ andb∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.2 Irreducible Brillouin Zone (BZ) . . . . . . . . . . . . . . . . . . . . . . . . . 142B.3 Two-dimensional Bravais (or space) lattices . . . . . . . . . . . . . . . . . . 143D.1 Conductor-backed folded slot with periodic dielectric substrate . . . . . . . 154D.2 Simulation unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155D.3 Unit cell dimensions for (a) square dielectric rods and (b) circular dielectric

rods where for the square rod, a is the unit cell size, b is the element edgelength, and for the circular rod, a is the unit cell size and b is the diameter 156

D.4 Band-gap plot as a function normalized insert size b/a . . . . . . . . . . . . 156D.5 Full band structure of a square lattice of dielectric columns as a function of

normalized insert size b/a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157D.6 Irreducible Brillouin Zone (BZ) . . . . . . . . . . . . . . . . . . . . . . . . . 157D.7 Simulated (HFSS) transmission spectra for Γ–X (left) and Γ–M (right) di-

rections and calculated (MoM) band diagram (center) of a two-dimensionalEBG with a=1.2 cm, b=4.8 mm, and εr=10.2 for TM polarization. . . . . . 158

D.8 Top view of conductor-backed folded slot design with periodic dielectric sub-strate (a=1.2 cm, b=4.8 mm, and εr=10.2) where slot dimension in paren-theses refers only to the reference slot design on an homogeneous substrate 159

D.9 Measured and simulated frequency response of conductor-backed slot withperiodic dielectric substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

D.10 Measured normalized E - and H -plane antenna patterns (co- and x-pol) ofreference slot at 9.7 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

D.11 Evanescent field structure inside EBG-backed substrate (left) and propagat-ing mode in homogeneously-filled parallel-plate (right) . . . . . . . . . . . . 163

D.12 Measured normalized E - and H -plane antenna patterns (co- and x-pol) ofconductor-backed slot with EBG substrate at 9.7 GHz . . . . . . . . . . . . 164

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D.13 Measured normalized E -plane antenna patterns (co-pol) of the reference slotand the slot backed with the periodic dielectric at 9.7 GHz . . . . . . . . . 164

D.14 Axis and pattern cut definitions of the measurement systems . . . . . . . . 166

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LIST OF APPENDICES

APPENDIXA Transmission Line Formulation for the One-Dimensional Periodic

Array of Dielectric Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B Bravais Lattices and the Brillouin Zone . . . . . . . . . . . . . . . . . 140C Impedance Matrix Elements for Two-Dimensional Periodic Struc-

tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144D Parallel-Plate Mode Reduction In Conductor-Backed Slots Using

Periodic Dielectric Substrates . . . . . . . . . . . . . . . . . . . . . . . 152E Review Of The Method Of Moments . . . . . . . . . . . . . . . . . . . 168

xiii

CHAPTER 1

Introduction

For many years, electromagnetic theorists were provided with two primary means of pre-

dicting electromagnetic phenomena: measurements and analytical solutions. Measurements

of electromagnetic systems provide the scientist/engineer with reams of data and a physical

intuition for the mechanisms involved in producing the response. However, the feasibility

of fabricating new devices for each design change can be time-consuming and, depending

on the architecture or application, expensive. These two limitations are often addressed by

using analytical techniques to predict electromagnetic behavior. Solution techniques includ-

ing asymptotic analysis, variational analysis, integral transforms, contour integration, and

perturbation theory are the staple of analytical electromagnetic prediction and design. Im-

provements in computer speed and memory helped to provide the framework for the rapid

development of computational electromagnetics (CEM) methods including, but not limited

to, the finite element (FE) method, the finite difference methods in both time (FDTD)

and frequency domains (FDFD), the transmission line matrix (TLM) method, and method

of moments (MoM) solutions. CEM techniques have added significantly to the toolboxes

of scientists and engineers alike. In fact, the hybridization of many of the aforementioned

techniques to solve large and complex problems may provide revolutionary new solution

capabilities for the near future.

With the recent advent of general purpose electromagnetics (EM) codes1, a third so-

lution method has been introduced to the technical community, simulation. One might

conclude that the availability of high performance commercial EM software has mitigated

the need to solve specific electromagnetic problems by writing original computer codes.

For example, the late availability of commercial codes that incorporate periodic boundary1An extensive repository of commercial general purpose computational electromagnetic codes and locally

free software is found at the Electromagnetics Library (EMLIB), http://emlib.jpl.nasa.gov.

1

conditions is a prime example of a recent simulation tool that has renewed the interest in

developing new circuit and antenna devices. Certainly, one of the advantages of the general

EM solver is its ability to solve a wide variety of problems. The disadvantage of the general

EM solver is its inability to address specific theoretical or numerical issues that can signifi-

cantly affect the accuracy of a solution. This is not to say that general EM solvers are not

extremely useful for design and have not revolutionized the way that electromagnetics re-

search is being performed. However, every CEM code writer will attest to the many nuances

and small singular changes that must be included in an original numerical code to achieve

maximum performance. Complex, real-world problems are not always amenable to gen-

eral purpose solutions and often require customized solutions and programs to solve them.

Computational electromagnetics which includes electromagnetic theory, sophisticated ana-

lytical techniques, and numerical methods is used to characterize and analyze the merits

and demerits of a new design.

The rapidly growing field of wireless communications is providing new opportunities to

develop novel structures that will enhance or even replace existing circuits and antennas.

Devices that incorporate periodicity as a key feature of the design are promising unri-

valed performance in microwave circuits and antennas. Periodic electromagnetic structures

are commonplace in many of the items we rely on every day – from the magnetron in a

microwave oven to the ultraviolet radiation protection provided by polarized sunglasses.

Commenting on the recent inundation of applications for periodic structures, Maddox [57]

remarks, “If only it were possible to make materials [photonic band-gap materials] in which

electromagnetic waves cannot propagate at certain frequencies, all kinds of almost-magical

things would happen.” It is important, nevertheless, to dispel some of the myths about

what photonic band-gap (PBG) materials are and are not. PBGs are periodic dielectric

and/or metallic structures that when designed and implemented correctly can improve the

performance of specific devices. PBGs are not magical structures that defy basic laws of

physics and have only of late appeared in the literature.

Photonic band-gap materials, sometimes referred to as electromagnetic band-gap (EBG)

materials, electromagnetic crystals (EC) or photonic crystals, are inhomogeneous structures

composed of periodic regions of material with a specific permittivity immersed in a homoge-

neous background of different permittivity. These artificial, composite structures have been

found to have unique properties that are advantageous in applications involving semicon-

ductor integrated circuits. Researchers from the diverse fields of classical electromagnetics,

solid-state physics, optics, material science, condensed matter physics, and semiconductor

2

physics, are actively contributing to the base of electromagnetic crystal knowledge. Active

areas of electromagnetic and photonic crystal research include but are not limited to mi-

crowave and millimeter-wave antenna structures, quasi-optical microwave arrays, photonic

crystal integrated circuits, high- and low-Q electromagnetic resonators, quantum optical

electromagnetic cavity effects, and optical nano-cavities. In addition, sonic band-gap ma-

terials or artificial acoustic crystal substrates, are being developed and could impact sonar.

From solid-state theory, we know that semiconductors allow electron conduction with-

out scattering only for electrons that have energies within a specific range of energy, often

termed band-gaps. Electromagnetic wave propagation in periodic dielectric media is analo-

gous to electron-wave propagation in semiconductor crystals. Although fundamentally dif-

ferent propagation mechanisms are involved, preliminary results suggest that at microwave

and millimeter-wave frequencies the propagation characteristics of these crystals can be ma-

nipulated by carefully designing and fabricating structures composed of regions of differing

dielectric constants. Early work by Yablonovitch [125] successfully demonstrated that light

propagation could be inhibited in certain frequency gaps in special photonic band-gap crys-

tals (PBGs). Scalar-wave-based theories were developed to determine a suitable candidate

for the first three-dimensional (3-D) photonic crystal. However, it became apparent that

scalar-wave-based solutions were inadequate for the task. Vector-wave-based solutions im-

plemented using the plane wave expansion (PWE) method for Maxwell’s equations, being

developed concurrently by Leung et al. [52], Zhang et al. [138], and Ho et al. [36], offered

the potential of finding a true photonic band-gap structure.2 Indeed, soon thereafter, a

structure that possesses a full photonic band-gap was identified. Theoretical work by Ho et

al. [36] confirmed the existence and structure of the first (3-D) photonic crystal, the di-

amond dielectric structure. Concurrently, band structures for two-dimensionally periodic

structures were being calculated in a similar manner by Plihal et al. [73]. However, it

was McCall et al. [60] who were the first to calculate and measure microwave propagation

and reflection in a two-dimensional (2-D) array of low-loss high-dielectric-constant cylin-

ders. With the confirmation of realizable 2-D and 3-D photonic crystal structures, attention

shifted to applying these materials in new designs and architectures.

Reviews of early photonic band-gap research can be found in special issues of the Journal

of the Optical Society of America B [1] and the Journal of Modern Optics [2]. Recently, a

book outlining photonic band theory and covering a wide range of photonic crystal applica-

tions was published by Joannopoulos [38]. Although previous research had focused almost2Earlier experimental work had only verified a pseudo-band-gap, not a true photonic band-gap. See [126].

3

exclusively on optical and quasi-optical applications for the PBG materials, it quickly be-

came apparent that numerous applications for similar periodic structures in the microwave

and infrared frequency range were being developed. Consequently, a joint special issue

of the IEEE Transactions on Microwave Theory and Techniques [3] and the Journal of

Lightwave Technology [4] was published to catalog the design, synthesis, and application of

electromagnetic crystal structures begin developed in the microwave and millimeter-wave

community. Concurrently, a special issue that focused on the particular theoretical and

numerical aspects of photonic band-gap structure research was released by the journal Elec-

tromagnetics [5].

It is the purpose of this work to analyze periodic structures to determine effective and re-

alizable uses for microwave circuits and antennas. In Chapter 2, full-wave integral equation

(IE)/method of moments (MoM) solutions are developed and implemented to determine the

band structure (propagating modes) of a periodic, one-dimensional inhomogeneous dielec-

tric region. A single electric field integral equation (EFIE) or coupled EFIEs incorporating

both the periodic free-space Green’s function (1-D) and equivalent electric polarization cur-

rents are formulated to determined the quantities of interest. Subsequently, the geometry

and integral equation(s) are discretized, and the method of moments is used to numerically

solve the resulting matrix equation. A nontrivial solution for the fields requires the matrix

determinant to be zero, which results in a characteristic equation. The eigenvalues (prop-

agation constants) are obtained from the roots of this equation. For a lossless structure,

the propagation constants of a guided wave are real numbers. However, in the stopbands,

the propagation constants are complex-valued. To validate the solution obtained through

the use of the IE/MoM approach, the solution of the exact eigenvalue equation for the

one-dimensional periodic problem is also obtained using the plane wave expansion (PWE)

method.

In Chapter 3, similar approaches as implemented in Chapter 2 are used to extend the

solution to two-dimensionally periodic media. Additional integral equation (IE)/method of

moments (MoM) codes are developed and implemented, but this time to determine the band

structure (propagating modes) of a periodic, two-dimensional inhomogeneous dielectric re-

gion. A single EFIE or coupled EFIEs are formulated using a periodic Green’s function

(2-D) along with equivalent electric polarization currents, and the method of moments is

again used to numerically solve the resulting matrix equation. The exact eigenvalue equa-

tion for the two-dimensional periodic problem is solved as before using the PWE method

to verify the IE/MoM solution. To simplify the computation of the band structure for two-

4

dimensionally periodic media, effective medium theory is applied in Section 3.4 to reduce

the two-dimensional periodic structure to a one-dimensional equivalent structure. This use-

ful technique can be applied when the period of the structure is much smaller than the

wavelength.

Recently, Ansoft, a leading developer of computational electromagnetic software, ex-

tended the capabilities of their 3-D commercial finite element software package to include

linked boundary conditions, their particular implementation of periodic boundary condi-

tions. The new capability enables Ansoft to perform new calculations on infinite arrays

similar to the waveguide simulator concept developed in the 1960s to determine array per-

formance without building an entire array assembly. A number of modeling methodologies

including direct-transmission methods, dispersion diagram methods, and reflection phase

analysis methods were examined to test the new linked boundary conditions [77]. To val-

idate the accuracy of the direct-transmission method, Ansoft chose to repeat the solution

developed in Chapter 3 and published by this author et al. in [91] to determine the band

structure (propagating modes) of a periodic, two-dimensional inhomogeneous dielectric re-

gion. The confidence in the accuracy of the full-wave integral equation (IE)/method of

moments (MoM) solution developed in this work provides a standard with which the new

linked boundary condition solution could be compared.

A number of researchers have designed electromagnetic crystal structures for use in

planar antenna and circuit applications, particularly for use as reflectors in planar dipole

antenna structures [18, 21, 17, 41, 95, 97]. Planar antennas are ideal for use in many

wireless networks including personal communications systems (PCS) and mobile satellite

communications. Microstrip antennas are of interest to many wireless users because of

their low aerodynamic profile (survivability/durability issue), light weight (volume/size),

and low cost. Microstrip antennas also have the advantage of being both conformal and

easily integrated into thin-film circuits. Unfortunately, microstrip antennas have relatively

low gain and narrow bandwidths due to substrate surface wave formation. By designing

special artificial substrates, surface wave formation may be reduced or even eliminated in

some planar antennas. This would dramatically increase both the available bandwidth and

gain to levels usually reserved for non-planar antennas. To this end, periodic structures

may serve as the material that can change the physical properties of substrates used in

fabricating planar circuits and antennas.

Only recently, have researchers developed periodic structures for application in slotted

antennas. Leung et al. [53] measured the radiation patterns of a slot antenna placed on a

5

layer-by-layer photonic band gap crystal. For planar antennas operating at a frequency in

the band-gap of the three-dimensional PBG crystal, energy which would have been radiated

into the substrate is reflected. However, at the interface between the PBG and the air,

the period of the PBG is broken and a parasitic mode (surface state) can exist. These

surface states decrease the efficiency by stripping power away from the radiating element.

However, by fabricating a resonant slot over a reflecting back plate and filling the resulting

parallel-plate with an appropriately designed artificial electromagnetic band-gap structure,

noticeable enhancements in both radiation pattern and bandwidth can be achieved using

a significantly lower profile than traditional designs. This design is detailed extensively in

Appendix D of this work.

As the need for very wideband, omni-directional antennas for use in mobile communica-

tion networks grows at an increasing rate, so does the need to develop custom solutions (and

the resulting codes) for novel antenna architectures and designs. The salient properties of

the antenna, such as its radiation pattern and efficiency, its input impedance, bandwidth

and gain, are all determined by the antenna’s electrical size, physical configuration, and the

environment in which it is located. In order to optimize the performance of the antenna, a

number of design parameters must be considered. Element shape, resonant frequency, gain,

and bandwidth of the individual element must be properly chosen. For planar circuits and

antennas, substrate characteristics must be designed that take into account the effect of

dispersion on the effective dielectric constant and the effect of substrate surface wave for-

mation. Other concerns, including the environmental effects of temperature, humidity, and

aging; the mechanical concerns of vibration effects and durability; and the ease of conforma-

bility/machinability must also be addressed. Additionally, feed structures must be designed

to minimize insertion loss, to maximize space utilization, and to minimize discontinuities.

Until recently [77], general EM solvers were unable to solve even simple periodic struc-

tures such as artificial dielectrics or large antenna arrays. Consequently, custom codes are

being developed to characterize important parameters such as the reflection coefficient, ra-

diation pattern, and radiation efficiency for planar antennas mounted on layered periodic

substrates. Significant attention has been directed to developing an entire class of structures

termed frequency selective surfaces [66]. A frequency selective surface is usually designed to

transmit or reflect electromagnetic radiation within a band of frequencies (or within specific

angular bands) from propagating through the surface. This is often accomplished by placing

metallic patch elements or aperture elements, such as annular rings, crossed slots, crossed

dipoles, or jerusalem crosses, in a periodic fashion on the surface of a dielectric layer. Ra-

6

diation near the resonant frequency of the element will be transmitted or reflected through

the surface as the design permits. Applications for such structures include broadband mi-

crowave antenna radome design, narrow-band frequency selective surfaces, and polarizers.

These structure are often evaluated using a generalized scattering matrix technique [135]

that cascades the propagation through the different layers. Less attention has been fo-

cused on three-dimensional frequency selective layers (volumes). The solutions developed

in Chapters 4 and 5 of this work are amenable to FSS design and have applications both

for substrate and superstrate design.

Recent numerical work in determining the microwave scattering from periodic material

implants in a layered medium has been done by Sarabandi [85], Sarabandi et al. [87],

Tsay et al. [114], Yang [130, 134], and Yang et al. [135, 136]. In Chapter 4, the solution

by Sarabandi [84] for the two-dimensional scattering from an inhomogeneous periodic layer

above a half-space layered medium is extended to general three-dimensional scattering from

an inhomogeneous doubly periodic layer above a half-space layered medium. The solution

of this problem is derived and implemented using a full-wave integral equation/method of

moments approach similar to the formulation for the one- and two-dimensional periodic

structures of Chapters 2 and 3.

Of particular concern in the solution of periodic structures is the convergence of the

resulting Floquet series. Even though the contribution of the off-plane periodic elements

converges very quickly, the convergence of the on-plane periodic elements is notoriously slow.

In order to compute the impedance matrix elements in a reasonable amount of time, various

series acceleration techniques and transformations have been suggested [42]. Summation

acceleration techniques convert a slowly converging series to a rapidly converging one by

allowing the series to be transformed into a second series that converges to the same limit but

does so in a rapid fashion. For the problems of interest in Chapter 4, the combination of a

Poisson transformation [70] and a Shanks’ transformation [90] are successfully implemented

to improve both the speed and the accuracy of the impedance matrix element computations.

This is the first known successful implementation of a Shanks’ transformation applied to

each of the series in the double sum found in the planar two-dimensional, periodic free-

space Green’s function. Additionally, Kummer’s method [51] is applied to specific series to

compare the convergence rate of these different series acceleration techniques. A complete

treatment of these and other series acceleration techniques is found in Section 4.5.

Of interest to the applied microwave community is the implementation of a microstrip

patch antenna on a doubly periodic dielectric layered medium. The solution procedure

7

developed in Chapter 4 to determine the scattering from doubly periodic dielectric layered

medium is used to validate an equivalent model that replaces the periodic layer with an

equivalent uniaxial layered medium. The ability of the model to emulate the periodic layer is

validated using plane wave reflection coefficients for various combinations of filling fractions,

angles, and permittivities and the limitations of the model are addressed. Subsequently, the

solution for a rectangular microstrip patch element radiating over the equivalent uniaxial

layered medium is carried out accurately and efficiently using a commercial finite element

package packages that incorporates anisotropic substrates. Conclusions about the effect of

the anisotropy on various antenna parameters such as resonant length, pattern shape, and

array coupling are drawn and numerical examples that illustrate the salient features of the

single patch antenna integrated on a uniaxial substrate are presented in Chapter 5.

Traditionally, microstrip patch antennas have been integrated on relatively low per-

mittivity substrates in order to improve antenna performance. Integrating the antenna

on higher permittivity substrates is preferred to minimize circuit size and spurious radia-

tion [75, 89] but at the cost of confining the potential radiating energy even more tightly.

This trade-off between good antenna performance and good circuit performance is a key

design feature found in many microstrip antenna designs. Although a good deal of atten-

tion has focused on integrating microstrip patches on homogeneous substrates, many of the

practical substrates with higher permittivities in use today such as sapphire have a signifi-

cant amount of (uniaxial) anisotropy. The primary effect of anisotropy on rectangular patch

antenna design is the change in its resonant length (frequency). This is significant because

of the narrow bandwidth of the patch itself. The relatively large shift in resonant frequency

produced in many of the modern substrates may actually force a rectangular patch de-

signed to operate at a specific frequency to radiate outside of the antenna bandwidth [74].

Additionally, anisotropic effects are found that shape the radiation pattern of the patch

and thus in an array configuration, the coupling to other elements. Because uniaxial sub-

strates are often expensive to manufacture and have limited flexibility for design, uniaxial

substrates can be emulated (and easily fabricated) by incorporating periodic inclusions in

an otherwise homogeneous substrate. The doubly periodic structure can be constructed us-

ing simple milling or etching techniques from simple inexpensive, homogeneous substrates.

This is significant because some common uniaxial materials such as sapphire that are ex-

pensive to grow [6] can be “artificially” replicated easily and inexpensively. Additionally,

the artificial nature of the periodic uniaxial substrate permits the creative design of new

substrates with the expanded freedom of anisotropic ratio, background permittivity, and/or

8

fabrication technique.

A number of potential applications for the solution techniques developed in this work

are outlined in Chapter 6. The specification of an arbitrarily shaped dielectric is one inter-

esting design feature that can be optimized in the solution of the one- and two-dimensional

periodic media. Additionally, lattices of dielectric elements with differing dielectric con-

stants can be included to provide additional freedom for the design. Two potential appli-

cations for effective medium theory that promise to be of value are the extension of EMT

to off-axis propagation in two-dimensional lattices and to out-of-plane propagation in two-

dimensional lattices. A number of new applications can be developed using the solution

technique implemented in Chapters 4 and 5 including new frequency selective surfaces (vol-

umes), incorporating separate periodicities (non-commensurate periodicities) for each layer,

incorporating material implants within each layer of differing relative permittivity (dielec-

tric and/or metallic loading), and the extension of the solution to large planar antenna

elements. Perhaps the most significant development would be incorporation of many of the

above suggestions to eliminate surface wave formation in planar antenna applications.

9

CHAPTER 2

One-Dimensional Periodic Dielectric Structures

2.1 Introduction

One-dimensional periodic electromagnetic structures are commonplace in many of the

items we rely on every day – from the magnetron in a microwave oven to the ultraviolet

(UV) radiation protection provided by polarized sunglasses. In particular, the field of optics

is replete with one-dimensional periodic structures. How periodicity is incorporated into

each device is what distinguishes one design from another. For example, the direction of

period in some structures is in the direction of stratification, whereas in others, the period

is in a transverse plane to the direction of stratification. Bragg mirrors (reflectors) are an

excellent example of periodicity in the direction of stratification. The highly reflective coat-

ings used in Bragg reflectors are produced by “sandwiching” layers of differing dielectric

material with highly reflective properties. A recent example of active research in this area

has been reported in [22] where total omni-directional reflections from one-dimensional

dielectric lattices have been reported. Other application in this class include specialized

thin-film waveguiding, quarter-wave stacks, and folded Solc filters [137]. Alternating layers

of metal and dielectric are being developed constantly that provide almost unbelievable

new applications. Transparent metallic structures have been developed that permit the

transmission of light over a tunable range of frequencies, effectively blocking both UV and

infrared and lower frequencies from propagating through [88]. Optical and quasi-optical

researchers have produced new sensor and eye protection devices, heat reflecting windows,

better ultraviolet blocking films, transparent electrodes for light emitting diodes, and light

crystal displays. The second class of periodicity, corrugation along the surface of a dielectric

medium, is used in devices such as diffraction gratings and high-reflectance distributed-

10

Bragg-reflector (DBR) lasers. Diffraction gratings are the most commonly known appli-

cation of corrugations and are used to provide certain spectral or angular characteristics

for reflection and transmission of electromagnetic radiation. Other interesting applications

include distributed-feedback (DFB) lasers where an active layer provides gain, TM-to-TE

mode conversion, and forward-backward mode conversion.

For microwave and RF engineers, the gridded traveling wave tube (GTWT) amplifier is

a prime example of harnessing the potential power of a periodic device. The GTWT uses

a helical waveguide to convert the high-energy of an electron beam to microwave energy

and is implemented on almost every airborne radar in the world. At lower frequencies, the

remote sensing community is actively studying naturally occurring one-dimensional periodic

structures such as traveling ocean waves and periodic geological structures and man-made

periodic structures such as periodic vegetation [112].

A fundamental understanding of how electromagnetic fields behave in a periodic medium

is required not only to apply these unique properties correctly but also to critique designs

or theories where they are applied incorrectly. The foundation for electromagnetic wave

propagation in periodic media is provided in Section 2.2. The phase constants for elec-

tromagnetic waves propagating in a one-dimensional lattice of dielectric slabs in an air

background is found explicitly using two different techniques. The first solution, detailed

in Section 2.3, incorporates the use of a Fourier series representation for the periodic field

and is determined by solving the one-dimensional wave (differential) equation for periodic

media. The second solution is found by deriving an integral equation for the periodic field

and numerically solving the resulting linear system and is presented in Section 2.4. A

number of representative structures are used in the chapter to show the behavior of elec-

tromagnetic fields in one-dimensional periodic media and observations about the salient

features of each are discussed. Although theoretical band structures are useful, actual re-

alizable devices that incorporate a design and that can be fabricated also provide valuable

insight for understanding the propagation mechanisms at work. An example solution for

the microstrip excitation of a one-dimensional periodic dielectric structure is presented in

Section 2.5. Since the fields produced by the quasi-TEM mode of the microstrip (for high

dielectric constant) are confined relatively close to the microstrip itself, the dielectric sub-

strate is modelled as a hi-Z, low-Z filter. The filter design is shown to match closely to

that of periodic plane wave propagation in a lattice of dielectric slabs and to the measured

response of the microstrip line itself.

11

2.2 Fundamental Concepts

2.2.1 Maxwell’s Equations

The fundamental equations that form the foundation for electromagnetic theory are

Maxwell’s equations. Written in differential form, these equations are

∇× E = −∂B∂t

(2.1a)

∇× H = J+∂D∂t

(2.1b)

∇ · D = ρ (2.1c)

∇ · B = 0 (2.1d)

where E is the electric field intensity, B is the magnetic flux density, H is the magnetic

field intensity, and D is the electric flux density. The electric current density J and electric

charge density ρ are the sources of the electromagnetic fields.

For linear and isotropic media, E and D and B and H are related by the constitutive

relations

D = εrε0E (2.2a)

B = µrµ0H, (2.2b)

where ε0, µ0, εr, and µr are the free-space permittivity, free-space permeability, relative

permittivity, and relative permeability, respectively.

The equation of continuity provides the relation between the electric charge density and

electric current density through

∇ · J = −∂ρ∂t

(2.3)

which is statement of the conservation of charge.

The remaining fundamental equation, Lorentz’s force equation, determines the total

electromagnetic force on a charge q to be

F = q (E + u× B) (2.4)

where u is the velocity of the moving charge.

The eight equations, comprised of the four Maxwell’s equations (2.1), together with the

constitutive relations (2.2), the equation of continuity (2.3), and Lorentz’s force equation

(2.4), provide the necessary framework to predict all macroscopic electromagnetic interac-

tions. Although the four equations (2.1a)–(2.1d) are not independent, equations (2.1c) and

(2.1d) can be derived from (2.1b) and (2.1a) using (2.3).

12

2.2.2 Propagation of Waves in a Periodic Medium

One of the many interesting problems in mathematical physics is the solution of the wave

equation in periodic media. The solution of this type of equation is usually derived using

a form of Floquet’s theorem. Floquet’s theorem, as it was originally presented [120], deter-

mined a particular periodic solution of Mathieu’s equation, the equation of wave motion.

Subsequently, this solution was extended for Hill’s equation, a generalization of Mathieu’s

equation, for any periodic function, providing the basis for one of the great successes in

applied physics: quantum band theory. Band theory, which describes the properties of

electrons in a periodic potential due to the atomic arrangement of atoms in a crystal, is

the foundation for the understanding of electronic transport in metals, semiconductors, and

insulators. The solution of the periodic potential problem can be expressed in mathematical

form as Bloch’s theorem. Bloch’s theorem states that the eigenfunctions of the Schrodinger

equation for a periodic potential are the product of a plane wave and a function which has

the same period as the periodic potential [98]. A number of similarities can be seen in the

solution of electron wave propagation in semiconductors and electromagnetic wave propa-

gation in periodic dielectric media, and Bloch’s theorem can be extended to electromagnetic

wave propagation in periodic media.

For propagating modes in a periodic dielectric structure, the field in an adjacent cell is

related by a complex constant. That is,

E(x, y, z + a) = E(x, y, z) e−jβ0a (2.5)

where E(x, y, z) is a periodic function of z with period a and β0 is the phase constant in

the z direction. If the propagating mode in the structure has the form

E(x, y, z) = Ep(x, y, z) e−jβ0z (2.6)

where Ep(x, y, z) is a periodic function of z with period a, then

E(x, y, z + a) = Ep(x, y, z + a) e−jβ0(z+a). (2.7)

Since Ep(x, y, z) is periodic with period a,

Ep(x, y, z + a) = Ep(x, y, z). (2.8)

Substituting (2.8) into (2.7) yields

E(x, y, z + a) = Ep(x, y, z) e−jβ0(z+a). (2.9)

13

But, using (2.6),

E(x, y, z + a) = E(x, y, z) e−jβ0a, (2.10)

which is exactly (2.5), a statement of Floquet’s (or Bloch’s) theorem.

From the theory of Fourier series, the periodic electric field with period a can be ex-

panded as a periodic function in x and prescribed phase constant β0 and is given by

E(x, y, z) =∑n

En(x, y) e−j2πn

az e−jβ0z =

∑n

En(x, y) e−jβnz (2.11)

where

En(x, y) =1a

a∫0

En(x, y, z) ej2πn

az dz

are the coefficients that serve to represent the dependence on x and y, and

βn = β0 +2πna

is the phase constant of the nth harmonic.

Thus, the field in a periodic structure can be expanded in an infinite set of harmonics

through Floquet’s theorem, each with frequency f and propagation constant βn. The phase

velocity νn of the nth harmonic is

νn =ω

βn=

ω

β0 + 2πna

. (2.12)

For slow-wave structures, the phase velocity of the nth harmonic can be less than the free-

space velocity, or νn < c. However, the group velocity, νg, found from

νg =1

∂βn/∂ω=

1∂(β0 + 2πn

a

)/∂ω

=∂ω

∂β0(2.13)

is independent of n.

2.3 Plane Wave Expansion Method

2.3.1 Analytical Techniques

A straight forward solution to the exact eigenvalue equation for the one-dimensional

periodic problem is obtained through the use of the Fourier series. A representative one-

dimensional periodic array of dielectric slabs with period a and dielectric insert width b is

14

x

y

ba

E(H)

φz

Figure 2.1: One dimensional lattice of dielectric slabs of width b in a periodic lattice withperiod a

illustrated in cross section in Figure 2.1. If the electric field does not have a component

in the x -direction (the direction of period or stratification), the mode is denoted TEx or

horizontal. If the magnetic field does not have a component in the x -direction, the mode is

denoted TMx or vertical. For normal incidence, the modes are degenerate (TEx is equivalent

to TMx).

Transverse Electric (TEx) Case

The electric field can be expanded as a periodic function of plane waves in x with period

a and prescribed propagation constant of kx0 and is given by

E(x, y) = zEz(x, y) = zEp(x) e−jkx0x e−jkyy (2.14)

where Ep(x) is the periodic electric field that propagates only in the xy-plane, i.e. kz = 0,

without loss of generality. Since the electric field must satisfy the wave equation, we now

apply the operator(∇2

xy + k2)to Ez(x, y) of (2.14) noting that the dielectric constant is a

function of x

∇2xyEz(x, y) + k

20εr(x)Ez(x, y) = 0. (2.15)

15

Assuming the parallel slabs are infinite in the y and z directions, (2.15) can be simplified

to

− d2

dx2Ez(x, y) + k2

yEz(x, y) = k20εr(x)Ez(x, y). (2.16)

The periodic electric field is expanded in a Fourier series in x with unknown coefficients an

which serve to represent the dependence on y

Ep(x) =∑n

an e−j 2πn

ax. (2.17)

Since the dielectric function is also periodic, it is appropriate to expand it in another Fourier

series with coefficients bm

εr(x) =∑m

bm e−j 2πm

ax. (2.18)

Substituting the Fourier expansions for the field and the dielectric into (2.16) and carrying

out the algebraic operations, we obtain

∑n

[(2πna

+ kx0

)2

+ k2y

]an e

−j 2πnax = k2

0

∑n

∑m

anbm e−j 2πm

ax e−j

2πnax. (2.19)

In order to determine the unknown coefficients an and bm, (2.19) is multiplied by an orthog-

onal function and integrated over one unit cell which produces a Kronecker delta function

for a specific index

∑n

[(2πna

+ kx0

)2

+ k2y

]anδ

(2πpa

− 2πna

)= k2

0

∑n

∑m

anbmδ

(2πpa

− 2πma

− 2πna

).

(2.20)

The convolution in equation (2.20) is easily cast into the following general matrix form[(2πna

+ kx0

)2

+ k2y

]an = k2

0

∑m

bn−mam (2.21)

where

bn−m =1a

b/2∫−b/2

(εr − 1) e−j2π(n−m)

ax dx+

1a

a/2∫−a/2

(1) e−j2π(n−m)

ax dx

=b

a(εr − 1) sinc

π(n−m)ba

+ δn−m. (2.22)

A generalized linear eigensystem problem is represented by Ax = λBx where A and B

are n×n matrices. The value λ is an eigenvalue and x = 0 is the corresponding eigenvector.

The propagating modes in the TEx case are solutions of the generalized linear eigensystem

in (2.21).

16

Transverse Magnetic (TMx) Case

The solution for the TMx case is similar to the TEx case with the significant difference

that the electric field in (2.14) is replaced by the magnetic field in the wave equation

∇×

1εr(x)

∇× zHz(x, y)+ k2

0zHz(x, y) = 0. (2.23)

In the derivation of (2.15), the curl operator is applied to both sides of (2.1a). The expression

for the magnetic field found from (2.1b) is then substituted into the resulting equation which

when simplified, yields (2.16). However, when the curl operator is applied to (2.1b) first,

as it is in the TMx case, the operator acts on both the field and the periodic dielectric

function. Careful attention must be directed to correctly evaluating the specified operators.

The resulting equation that must be solved is

1εr(x)

z · ∇ ×∇× zHz(x, y) + z · ∇

1εr(x)

×∇× zHz(x, y) = −k2

0Hz(x, y). (2.24)

The Fourier series representation of the periodic magnetic field with unknown coefficients

an is

Hp(x) =∑n

an e−j 2πn

ax (2.25)

and for the inverse of the dielectric function, it is

1εr(x)

=∑m

bm e−j 2πm

ax (2.26)

with coefficients bm different from coefficients bm in (2.18) for the TEx case. By substituting

the expressions for the field and dielectric function into (2.24), carrying out the curl and

gradient operations, and simplifying the resulting expression yields

∑m

bme−j 2πm

ax

− d2

dx2

∑n

an e−j 2πn

ax e−jkx0x + k2

y

∑n

an e−j 2πn

ax e−jkx0x

− d

dx

∑m

bme−j 2πm

ax d

dx

∑n

an e−j 2πn

ax e−jkx0x = −k2

0

∑n

an e−j 2πn

ax e−jkx0x. (2.27)

If the derivatives in (2.27) are evaluated and integrated over one unit cell, then the resulting

expression can be cast into a different matrix equation to solve for the unknown eigenvalues

∑m

bn−mam

[(2πna

+ kx0

)2

+ k2y −

2π(n−m)a

(2πna

+ kx0

)]= −k2

0an (2.28)

17

where

bn−m =1a

b/2∫−b/2

(1εr

− 1)e−j

2π(n−m)a

x dx+1a

a/2∫−a/2

(1) e−j2π(n−m)

ax dx

=b

a

(1εr

− 1)sinc

π(n−m)ba

+ δn−m. (2.29)

An ordinary linear eigensystem problem is represented by the equation Ax = λx where

A denotes an n × n matrix. The propagating modes in the TMx case are solutions of the

ordinary eigensystem problem in (2.28).

2.3.2 Matrix Solution of the Eigensystem

The solution of (2.21) or (2.28) is amenable to fast computation using the commercial

software package MATLAB. The resulting eigenvalues of the matrix are the squares of the

frequencies of the propagating modes in the structure. The solution of the frequencies of

the propagating modes in the structure are found for specific values of phase shift kx0a ∈[0, 2π]. Of course, if an infinite number of Floquet modes are used in the solution, then an

infinite number of frequencies (space harmonics) will satisfy (2.21) or (2.28). However, the

limitation of a finite computer memory requires that only a finite number of frequencies

can be found for a given phase shift. If the resulting eigenvalues are sorted from largest

to smallest, the last eigenvalue yields the lowest space harmonic that will propagate for a

given phase shift. Additional propagating modes are found from the ascending eigenvalues.

In Figure 2.2 is plotted a finite portion of the Brillouin zone (BZ) diagram1 for a normally

incident TEx mode (ky=0) in a periodic array of dielectric slabs with filling fraction b/a=0.1

and relative dielectric constant εr=8.9. The filling fraction is defined as the volumetric

ratio of dielectric material to the total volume. The band structure is shown for normalized

propagation values kx0a ∈ [−3π, 3π]. The dark regions are the stop bands. Seeing that

the diagram repeats itself every 2π, only normalized propagation values between 0 and π

need to be calculated to determine all of the propagating modes in the structure. This

region is often termed the irreducible Brillouin zone [38, 44]. For one-dimensional periodic

structures, this a trivial determination. For the two-dimensional structures encountered in

Chapter 3, the irreducible Brillouin zone becomes more complicated. A complete treatment

of the Brillouin zone for two-dimensional lattices is given in Appendix B.1For band structures in the literature, the normalized propagation constant is plotted on the abscissa

and the normalized frequency is usually plotted on the ordinate.

18

Normalized propagation constant (k xa)

-9.42 -6.28 -3.14 0.00 3.14 6.28 9.42

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.5

1.0

1.5

2.0

2.5

Figure 2.2: Full band structure for a normally incident mode in a periodic array of dielectricslabs with b/a=0.1 and εr=8.9

The width of the stopbands and passbands is a function of the relative dielectric con-

stant εr and the filling fraction b/a of the one-dimensional lattice. To illustrate this fact,

representative band structures for a one dimensional lattice of dielectric slabs with filling

fraction b/a=0.5 for various combinations of dielectric contrast are shown in Figure 2.3

for kx0a ∈ [−π, π]. When the permittivity difference between the two regions is negligible

(Figure 2.3(a)), the wave propagates uninhibited for all frequencies, i.e., no stopgaps. If a

small dielectric contrast is introduced between the two materials (Figure 2.3(b)), small gaps

can form. However, the required interference effect between the periods is simply too small

to produce a gap of any consequence. Large contrasts in dielectric constant can produce

significant gaps as seen in Figure 2.3(c).

A number of simple checks can be performed to determine whether the solution for

the periodic slabs is correct in addition to checking the calculated values against published

data. For convenience, the slab is placed in the center of the unit cell and the propagation

constant is determined for a specific combination of electrical and geometrical parameters.

However, the particular location of the slab within the unit cell should have no effect on

the propagating structure. In particular, this is found to be true for two small strips of

dielectric material each half the original dielectric insert width located at the edges of the

unit cell. Other observations are drawn from the calculations by reducing the geometrical

and electrical parameters to special cases, including the following: (i.) as the filling frac-

19

Normalized propagation constant (k xa)

-3.14 -1.57 0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30εb=13,εa=13

(a)

Normalized propagation constant (k xa)

-3.14 -1.57 0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30εb=12,εa=13

(b)

Normalized propagation constant (k xa)

-3.14 -1.57 0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

εb=1,εa=13

(c)

Figure 2.3: Band structure for one dimensional lattice of dielectric slabs of with fillingfraction b/a=0.5 for (a) εb=εa=13, (b) εb=12, εa=13, and (c) εb=1, εa=13

tion decreases to zero, the calculated propagation constant kx0 approaches the minimum

value, the free-space propagation constant k0 (Figure 2.4); (ii.) as the filling fraction in-

creases to 1, the calculated propagation constant approaches the maximum value, k0√εr

(Figure 2.4); and (iii.) as the relative dielectric constant εr is reduced to 1, the calculated

propagation constant approaches the minimum value, the free-space propagation constant

k0 (Figure 2.5).

In Figure 2.6, the band structure (TMx) for the lowest order mode of a periodic array

of dielectric slabs with filling fraction b/a=0.3545 and εr=8.9 is shown for different values

of off-axis propagation. The values for the filling fraction and the relative dielectric con-

stant are obtained from a two-dimensional structure in [64, 129]. In order to determine the

correctness of the results obtained for the two-dimensional periodic structure, the filling

fraction in one lattice direction is allowed to increase until an equivalent one-dimensional

20

Normalized propagation constant (k xa)

0.00 0.79 1.57 2.36 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.1

0.2

0.3

0.4

0.5b/a=0.05b/a=0.1b/a=0.25b/a=0.5b/a=0.75k0

k0εr1/2

Figure 2.4: Band structure (TMx) for lowest order mode of a periodic array of dielectricslabs as a function of filling fraction b/a for normal incidence and εr=8.9

Normalized propagation constant (k xa)

0.00 0.79 1.57 2.36 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.1

0.2

0.3

0.4

0.5er=2

εr=4

εr=9

εr=16

εr=25

k0

Figure 2.5: Band structure (TMx) for lowest order mode of a periodic array of dielectricslabs as a function of dielectric constant εr for normal incidence and fillingfraction b/a=0.3545

periodic structure is obtained. Thus, the solution of the one-dimensional equivalent struc-

ture is needed. Similar results for the band structure of the lowest order TEx mode are

not shown. Note that for normal incidence (ky0 = 0), the first (and lowest) band does not

open up until the normalized frequency f0 ≥ 0.2. However, when the mode is allowed to

propagate with an off-axis component, i.e. φ0 = 0), the lowest band opens up for small

21

normalized frequencies. As the off-axis propagation constant increases, the bandwidth of

the lowest stopband increases and the bandwidth of the lowest passband decreases. In the

limit as the off-axis propagation increase, the passband becomes effectively flat over a small

range of frequencies that correspond to discrete propagating modes in a inhomogeneously-

filled parallel plate waveguide [55] and is also found to hold for two-dimensional structures

(waveguides) as well [96].

Normalized propagation constant (k xa)

0.00 0.79 1.57 2.36 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ky=0

ky=1/a

ky=1.5/a

ky=2/a

ky=2.5/a

Figure 2.6: Band structure (TMx) for lowest order mode of a periodic array of dielec-tric slabs as a function of off-axis propagation constant ky0 for filling fractionb/a=0.3545 and εr=8.9

Another check to determine if the code is calculating the correct propagation constant

is found by setting the filling fraction equal to some value x between zero (0) and one (1)

and obtaining a solution for the propagation constant. Subsequently, the filling fraction is

set equal to (1− x), the quantity (εr − 1) is replaced by (1− εr) in (2.21) and (2.25), and a

new solution is calculated. The solutions should be and are found to be exactly the same.

Asymptotic solutions are another effective way to check the solution. For small fill-

ing fractions, the propagation constant kx0 approaches k0√εeff where εeff is the effective

dielectric constant given by the volumetric average of the constitutive phases [24]

εeff =(1 + (εr − 1)

b

a

).

For a dielectric slab (εr=11) with a filling fraction of 0.1, the effective dielectric constant εeff

is estimated to be 2, corresponding to a propagation constant kx0 1.4k0. This approxima-

tion is only accurate to within 1% for relatively low normalized frequencies f0=fa/c < 0.1.

22

2.4 Formulation of Integral Equations

In this section, volume integral equations (IE) are derived from Maxwell’s equations

(2.1a)–(2.1d) and the boundary conditions, that when solved, yield propagation constants

from which the propagating mode structure can be determined. For the inhomogeneous

problems found throughout this work, integral equations are derived that incorporate equiv-

alent volume polarization currents.2 The integral equations are discretized and cast into a

matrix equation form through the use of the method of moments (MoM) numerical solu-

tion technique. The unknown is obtained from the solution of the resulting linear system.

For the problems solved in Chapters 2 and 3, the solution of the linear system returns the

eigenvalues (propagation constants) of the structure.

2.4.1 Derivation of Electric Field Integral Equation

In many scattering problems, the total electric field can be viewed as the sum of an

incident field Ei(r) due to radiation from a known source with the dielectric absent and a

scattered field E [J; r;V ] which is due to radiation by equivalent volume equivalent currents

J in a volume V

E(r) = E [J; r;V ] + Ei(r), (2.30)

in which the operator E [J; r;V ] can be expressed in terms of a Hertz potential [108] as

E [J; r;V ] = k20Π(r) +∇∇ · Π(r) (2.31a)

with

Π(r) =Z0

jk0

∫∫∫V

J(r′)e−jk0|r−r′|

4π|r − r′| dr′ (2.31b)

where k0 = ω√µ0ε0 is the free-space wavenumber of the background, ω is the frequency

of operation, Z0 =√µ0/ε0 is the intrinsic impedance of free-space, and V represents the

volume in which the sources reside. The field must satisfy

E [J; r;V ] + Ei(r) = E(r), r ∈ V. (2.32)

2In some homogeneous problems, integral equations are derived using equivalent surface currents reducingthe complexity and computational cost of determining the solution.

23

2.4.2 Equivalent Polarization Currents

The dielectric flux density D inside a dielectric immersed in a background medium of

free-space permittivity ε0 is equal to

D = ε0E + P (2.33)

where the polarization P is the electric dipole moment per unit volume. The polarization

current J is the time rate of change of the electric dipole moment per unit volume

J =∂

∂tP. (2.34)

Relating the polarization current to the difference between the induced electric flux density

and the electric flux density inside the dielectric through

J =∂

∂t(D − ε0E) (2.35)

yields the following expression for linear media

J =∂

∂t(εE − ε0E) . (2.36)

For time harmonic fields (ejωt), the equivalent polarization currents are related to the total

electric field by

J = jω (ε− ε0)E= jk0Y0 (εr − 1)E (2.37)

where Y0 = 1/Z0 is the intrinsic admittance of free-space.

TEx Case

Consider the representative dielectric slab (period not shown) in Figure 2.7 with period

a and dielectric insert width b. The slabs are excited by an incident time harmonic (ejωt)

plane wave with its electric field perpendicular to the plane of incidence (x-y plane) which

induces an electric current J with a z component only. Note that, since all quantities are

z invariant and because the induced electric current is z directed, ∇′ · J(r′) of (2.31a) is

zero. The total electric field everywhere is computed as the sum of the scattered field

produced by the equivalent induced electric current and the incident electric field given by

Ei = zE0e−jk0(x cosφ0+y sinφ0), which impinges on the slab in the k = (x cosφ0 + y sinφ0)

direction defined by an angle φi with respect to the x axis (φ0=0). This mode is denoted

24

y

xz

ba

E(H)

φ

Figure 2.7: Representative dielectric slab of width b with period a

as transverse electric to x, TEx, since for all angles of incidence the electric field does not

have a component in the x direction, the direction of stratification.

The dielectric material is replaced with equivalent volume currents using (2.37). For the

TEx case (assuming kz=0 without loss of generality),

Jz(x) = jk0Y0 (εr(x)− 1)Ez(x) (2.38)

where εr(x) is the permittivity function of the slabs within a unit cell given by

ε(x) =

εr, −b/2 < x < b/2;1, otherwise.

(2.39)

Since the dielectric material is periodic in x with period a, the resulting equivalent currents

must satisfy

Jz(x+ pa) = Jz(x)e−jkx0pa (2.40)

for a prescribed phase shift kx0a in the x direction. The scattered field is determined from

(2.31) by incorporating the periodic equivalent currents (2.40)

Esz(x) = −jk0Z0

b2∫

− b2

Jz(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′ (2.41)

where p is the Floquet mode index. Using Ez(x) = Esz(x) + E

iz(x), one can formulate the

25

following electric field integral equation (EFIE) to determine the equivalent currents

jk0Z0

b2∫

− b2

Jz(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′ +Jz(x)

jk0Y0(εr(x)− 1)= Ei

z(x). (2.42)

One-dimensional periodic free-space Green’s function The spatial form of the one-

dimensional periodic free-space Green’s function can be easily written in a more convenient

form through the use of the Poisson sum formula [70]. From Table I of [51],

∑p

f(p) =∑p

12jk

e−jk|x−pa|

=∑p

1a

[(2πpa

)2

− k2

]−1

e−j2πpax

= −1a

∑p

1[k2 −

(2πpa

)2]e−j 2πp

ax

(2.43)

where a is the unit cell width, k is the wavenumber of the medium, and p is the Floquet

index. Equation (2.43) becomes, upon substitution of a phase shift kx0 ,

∑p

e−jkx0pa1

2jke−jk|x−pa| = −1

a

∑p

e−jkxpx

k2 − k2xp

(2.44)

where

kxp =2πpa

+ kx0 .

Thus, (2.42) becomes upon substitution

−jk0Z0

a

b2∫

− b2

Jz(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ +Jz(x)

jk0Y0(εr(x)− 1)= Ei

z(x). (2.45)

TMx Case

Similarly for the the TMx case, the dielectric material is replaced with equivalent volume

currents. However, since the slabs are excited by an incident plane wave with its electric

field parallel to the plane of incidence, the induced electric current J can have both x and

y components. The total electric field everywhere is computed as the sum of the scattered

field produced by the equivalent induced electric current and the incident electric field given

by Ei = (z× k)E0e−jk0(x cosφi+y sinφi) which impinges on the slab in the k direction with φi

26

defined eariler. This mode is denoted as transverse magnetic to x, TMx, since for all angles

of incidence (assuming kz=0 without loss of generality), the magnetic field does not have a

component in the x direction.

For the TMx case, the dielectric material is replaced with equivalent volume currents

for both the x and y components

Jx(x) = jk0Y0(εr − 1)Ex(x) (2.46a)

Jy(x) = jk0Y0(εr − 1)Ey(x). (2.46b)

Using the periodic form of (2.46) in (2.31), the scattered field is found by integrating over

the induced current

Esx(x) = −jk0Z0

(1 +

1k20

∂2

∂x2

) b2∫

− b2

Jx(x′)∑p

e−jkx0pae−j

√k20−k2

y |x−x′−pa|

2j√k20 − k2

y

dx′

− kyZ0

k0

∂

∂x

b2∫

− b2

Jy(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′ (2.47a)

Esy(x) = −kyZ0

k0

∂

∂x

b2∫

− b2

Jx(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′

− jk0Z0

(1− k2

y

k20

) b2∫

− b2

Jy(x′)∑p

e−jkx0pae−j

√k20−k2

y |x−x′−pa|

2j√k20 − k2

y

dx′. (2.47b)

Using E(x) = Es(x) + Ei(x), one can formulate the following coupled EFIEs to determine

the equivalent currents

jk0Z0

(1 +

1k20

∂2

∂x2

) b2∫

− b2

Jx(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′

+kyZ0

k0

∂

∂x

b2∫

− b2

Jy(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′ +Jx(x)

jk0Y0(εr(x)− 1)= Ei

x(x)

(2.48a)

27

jk0Z0

(1− k2

y

k20

) b2∫

− b2

Jy(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′

+kyZ0

k0

∂

∂x

b2∫

− b2

Jx(x′)∑p

e−jkx0pae−j

√k20−k2

y|x−x′−pa|

2j√k20 − k2

y

dx′ +Jy(x)

jk0Y0(εr(x)− 1)= Ei

y(x).

(2.48b)

Using the Poisson sum formula (2.44) shown on page 26, and carrying out the derivatives

in (2.48) yields the coupled equations necessary to solve for the unknown equivalent currents

Jx and Jy,

− jk0Z0

a

(1− k2

xp

k20

) b2∫

− b2

Jx(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′

− kykxpZ0

jk0a

b2∫

− b2

Jy(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ +Jx(x)

jk0Y0(εr(x)− 1)= Ei

x(x) (2.49a)

− jk0Z0

a

(1− k2

y

k20

) b2∫

− b2

Jy(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′

− kykxpZ0

jk0a

b2∫

− b2

Jx(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ +Jy(x)

jk0Y0(εr(x)− 1)= Ei

y(x). (2.49b)

For a normally incident plane wave excitation, i.e. φi=0, the modes are degenerate (TEx

is equivalent to TMx).

2.4.3 Method of Moments Formulation

The first step in the method of moments (MoM) numerical method [34, 119] is to

discretize the geometry and approximate the unknown electric current in the dielectric

region with either subsectional or entire domain basis functions. The equivalent polarization

currents are expanded in a linear combination of these basis functions, and the equations

are tested in order to obtain an adequate number of equations to solve for the unknown

coefficients of the basis functions. In order to provide verification of the IE/MoM procedure,

computer codes have been written in Fortran to determine the propagating modes of a

periodic array of dielectric slabs.

28

Π Π

Π

x1

12

N

x2

xN∆x

Π3

(a) Piecewise constant basis functions

Λ 1Λ 2

ΛΝΛ 3

x2

x3

x1

xN∆x

(b) Piecewise linear basis functions

Figure 2.8: Piecewise constant and piecewise linear basis functions used in this work

TEx Case

Piecewise constant expansion / Piecewise constant testing The dielectric region

is discretized into N piecewise constant subsections, commonly referred to as pulse basis

functions, of width ∆x = b/N shown in Figure 2.8(a). The polarization current Jz is

approximated by a linear combination ofN piecewise constant basis functions with unknown

current coefficients Jzn located at the centers of the piecewise constant segments xn =

− b2 +∆x(n− 1

2),

Jz =N∑n=1

JznΠn(x) (2.50)

where

Πn(x) =

1, xn − ∆x

2 < x < xn + ∆x2 ;

0, otherwise.(2.51)

29

Inserting (2.50) into (2.42), yields

N∑n=1

Jzn

jk0Z0

a

xn+∆x2∫

xn−∆x2

∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ − Πn(x)jk0Y0 (εr − 1)

= Ei

z(x). (2.52)

Testing (2.52) with piecewise constant functions3 located at xm, and setting the excitation

to zero, yields the following equation

N∑n=1

Jzn

jk0Z0

a

xm+∆x2∫

xm−∆x2

xn+∆x2∫

xn−∆x2

∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ dx−xm+∆x

2∫xm−∆x

2

Πm(x)Πn(x)jk0Y0 (εr − 1)

dx

= 0, m = 1, 2, . . . , N (2.53)

which can be written in matrix form as[Zmn + δZ

] [Jzn

]=[0], (2.54)

where [Jzn ] is a column vector containing the unknown coefficients and [Zmn + δZ] is a

matrix whose elements are found by carrying out the integrations in (2.53). Specifically,

Zmn =jk0Z0∆2

x

a

∑p

sinc2(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

y − k2xp

(2.55a)

δZ = − δmn∆x

jk0Y0 (εr − 1). (2.55b)

The transforms of the piecewise constant testing and expansion functions, denoted Πm(kx)

and Πn(kx), are

Πm(kx) = ∆ sinc(kx∆2

)e−jkxxm (2.56a)

Πn(kx) = ∆ sinc(kx∆2

)ejkxxn . (2.56b)

Piecewise linear expansion / Piecewise linear testing Using piecewise linear func-

tions, commonly referred to as triangle basis functions, the dielectric region is discretized

into N subdomains of width ∆ = b/(N−1) that approximate the dielectric as shown in Fig-

ure 2.8(b). The polarization current Jz is approximated with a linear combination of these

N basis functions with unknown current coefficients Jzn centered at xn = − b2 +∆x(n−1),

Jz =N∑n=1

JznΛn(x) (2.57)

3When the expansion and testing functions are the same, the testing procedure is referred to as theGalerkin method [34, 83].

30

where

Λn(x) =

1− |x−xn|

∆x, xn −∆x < x < xn +∆x;

0, otherwise.(2.58)

Inserting the piecewise linear approximation for the current into (2.45) and testing the

resulting equation with piecewise linear basis functions, the elements of the impedance

matrix [Zmn + δZ] in (2.54) are found to be

Zm1 =jk0Z0∆2

x

a

∑p

(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−x1)

k20 − k2

y − k2xp

(2.59a)

Zmn =jk0Z0∆2

x

a

∑p

sinc4(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

y − k2xp

(2.59b)

ZmN =jk0Z0∆2

x

a

∑p

(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−xN )

k20 − k2

y − k2xp

(2.59c)

δZ = − δmn∆x

jk0Y0 (εr − 1). (2.59d)

The transforms of the piecewise linear testing and expansion basis functions, denoted

Λm(kx) and Λn(kx), are

Λm(kx) = ∆ sinc2(kx∆2

)e−jkxxm (2.60a)

Λn(kx) = ∆ sinc2(kx∆2

)ejkxxn . (2.60b)

The boundary condition at the edge of the dielectric insert requires continuity of the tan-

gential electric field. When using linear elements, this requires incorporating the two “half-

basis” functions, denoted Λ1(x) and ΛN (x), in Figure 2.8(b). The transforms of the two

“half-basis” functions in (2.59), Λ1 and ΛN , are

Λ1(kx) = ∆ejkxx1

(kx∆)2(1 + jkx∆− ejkx∆

)(2.61a)

ΛN (kx) = ∆ejkxxN

(kx∆)2(1− jkx∆− e−jkx∆

). (2.61b)

TMx Case

Piecewise linear expansion / Piecewise linear testing Care must be taken to cor-

rectly evaluate (2.49) because of the noticeable derivatives. The dielectric region is dis-

cretized into N subsections of width ∆ = b/(N − 1). The equivalent current J = xJx+ yJy

31

is expanded with a linear combination of N piecewise linear functions basis functions with

unknown current coefficients Jxn , Jyn centered at xn = − b2 +∆x(n− 1),

Jx =N∑n=1

JxnΛn(x) (2.62a)

Jy =N∑n=1

JynΛn(x) (2.62b)

with Λn(x) given in (2.58). Inserting (2.62) into (2.49) and testing the resulting equation

with piecewise linear functions yields

N∑n=1

Jxn

−jk0Z0

a

(1− k2

xp

k20

) xm+∆∫xm−∆

Λm(x)

xn+∆∫xn−∆

Λn(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ dx

+

xm+∆∫xm−∆

Λm(x)Λn(x)jk0Y0(εr(x)− 1)

dx

− Jyn

kykxpZ0

jk0a

xm+∆∫xm−∆

Λm(x)

xn+∆∫xn−∆

Λn(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ dx

= 0, m = 1, 2, . . . , N. (2.63a)

N∑n=1

Jxn

−kykxpZ0

jk0a

xm+∆∫xm−∆

Λm(x)

xn+∆∫xn−∆

Λn(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ dx

− Jyn

jk0Z0

a

(1− k2

y

k20

) xm+∆∫xm−∆

Λm(x)

xn+∆∫xn−∆

Λn(x′)∑p

e−jkxp(x−x′)

k20 − k2

y − k2xp

dx′ dx

+

xm+∆∫xm−∆

Λm(x)Λn(x)jk0Y0(εr(x)− 1)

dx

= 0, m = 1, 2, . . . , N. (2.63b)

Equation (2.63) can be written in matrix form asZxxmn Zxymn

Zyxmn Zyymn

+

δZxxmm 0

0 δZyymm

Jxn

Jyn

=

[0], (2.64)

32

where [Jxn Jyn ] is a column vector containing the unknown coefficients and [Zmn+ δZ] is a

matrix whose elements are found by carrying out the integrations in (2.63) yielding

Zm1xx =

jk0Z0∆2x

a

(1− k2

xp

k20

)∑p

(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−x1)

k20 − k2

xp

(2.65a)

Zmnxx =jk0Z0∆2

x

a

(1− k2

xp

k20

)∑p

sinc4(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.65b)

ZmNxx =jk0Z0∆2

x

a

(1− k2

xp

k20

)∑p

(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−xN )

k20 − k2

xp

(2.65c)

Zm1xy =

kykxpZ0∆2x

jk0a

∑p

(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−x1)

k20 − k2

xp

(2.65d)

Zmnxy =kykxpZ0∆2

x

jk0a

∑p

sinc4(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.65e)

ZmNxy =kykxpZ0∆2

x

jk0a

∑p

(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−xN )

k20 − k2

xp

(2.65f)

Zyx =Zxy (2.65g)

Zm1yy =

jk0Z0∆2x

a

(1− k2

y

k20

)∑p

(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−x1)

k20 − k2

xp

(2.65h)

Zmnyy =jk0Z0∆2

x

a

(1− k2

y

k20

)∑p

sinc4(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.65i)

ZmNyy =jk0Z0∆2

x

a

(1− k2

y

k20

)∑p

(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2 sinc2(kxp∆x

2

)e−jkxp(xm−xN )

k20 − k2

xp

(2.65j)

δZmmxx =− δmn∆x

jk0Y0 (εr − 1)(2.65k)

δZmmyy =− δmn∆y

jk0Y0 (εr − 1)(2.65l)

Piecewise constant expansion / Piecewise constant testing The solution can also

be carried out for the TMx case using piecewise constant expansion and testing functions.

The dielectric region is discretized into N subsections of width ∆ = b/N . The current

J = xJx + yJy is expanded in a linear combination of N piecewise constant basis functions

33

with unknown current coefficients Jxn , Jyn centered at xn = − b2 +∆x(n− 1

2 ),

Jx =N∑n=1

JxnΠn(x) (2.66a)

Jy =N∑n=1

JynΠn(x) (2.66b)

with Πn(x) given in (2.51). Inserting (2.66) into (2.49) and testing the resulting equation

with piecewise constant functions, the elements of the impedance matrix [Zmn + δZ] in

(2.64) are found to be

Zmnxx =jk0Z0∆2

x

a

(1− k2

xp

k20

)∑p

sinc2(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.67a)

Zmnxy =kykxpZ0∆2

x

jk0a

∑p

sinc2(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.67b)

Zmnyx =Zxy (2.67c)

Zmnyy =jk0Z0∆2

x

a

(1− k2

y

k20

)∑p

sinc2(kxp∆x

2

)e−jkxp(xm−xn)

k20 − k2

xp

(2.67d)

δZmmxx =− δmn∆x

jk0Y0 (εr − 1)(2.67e)

δZmmyy =− δmn∆y

jk0Y0 (εr − 1)(2.67f)

2.4.4 Matrix Solution

In order to accurately determine the eigenvalues of the impedance matrices given in

(2.55), (2.59), (2.65), and (2.67), a sufficient number of Floquet modes Np and subsectional

unknowns Nx must be included. A simple convergence criterion can be established to de-

termine when the computation of the individual impedance matrix elements has converged

adequately to its final value. A nontrivial solution for the fields requires the determinant

of the impedance matrix to be zero, which results in a characteristic equation. The eigen-

values (propagation constants) are obtained from the roots of this equation. For a lossless

structure, the propagation constants of a guided wave is a real number. However, in the

stopbands, the propagation constant is complex-valued.

The eigenvalues are calculated by minimizing a function of N variables subject to bounds

on the variables using a direct search algorithm. The algorithm uses the complex method

to find a local minimum point of a function of N variables and is based on function compar-

ison [139]. No derivative information is used in this technique. The minimization procedure

34

is carried out by fixing the frequency of operation and allowing the optimization code to

vary the propagation constant kx0 until a minimum is found. If more than one minimum

(eigenvalue) is found, implying that multiple modes are propagating in the structure, the

corresponding eigenvectors are analyzed to determine which eigenvalue corresponds to which

propagating mode. In the following tables, the propagation constant of the lowest order

mode that propagates is listed.

Table 2.1: Normalized TEx propagation constant kx0a for increasing number of Floquetmodes for f0=fa/c=1, εr=8.9, b=0.3545a, and Nx=40

kx0a (pulse) kx0a (triangle)

Np φ0=0 φ0=9.15 φ0=0 φ0=9.15

7 2.471 2.547 2.474 2.550

11 1.723 1.789 1.733 1.799

21 1.691 1.758 1.702 1.768

31 1.688 1.754 1.699 1.766

41 1.687 1.754 1.698 1.765

51 1.687 1.753 1.698 1.765

kx0a∗ 1.676 1.743 1.676 1.743

In Table 2.1, the normalized propagation constant kx0a for a normally incident plane

wave (φ0=0) and off-axis incident plane wave (φ0=9.15) for the TEx case is shown as

function of the number of Floquet modes (Np) and unknown basis set (pulse or triangle)

for the combination of normalized frequency f0=fa/c=1, 40 subsectional unknowns (Nx),

relative permittivity εr=8.9, and filling fraction b/a=0.3545. The value of φ0=9.15 cor-

responds to an off-axis propagation constant of ky0=1/a which is equal to the inverse of

the unit cell size. The off-axis propagation constant is set to this value because many au-

thors use this specific parameter value as a reference data set [38, 129]. The bold values

(error less than 1%) for Np=51 and Nx=40 listed in Tables 2.1–2.4 correspond to the same

solution and serve to represent a fixed parameter set to compare the relative convergence

as a function of number of unknowns and Floquet mode contributions. The correct value,

denoted kx0a∗ and listed at the bottom of Tables 2.1–2.4, is determined from a solution of

the exact eigenvalue equation derived earlier in the chapter.

Note in Table 2.1 that the necessary convergence occurs for both the pulse and triangle

basis cases for normal incidence and off-axis propagation as the number of Floquet modes

35

increase. The accuracy of the eigenvalue solution is limited by both Nx and Np. The

advantage of expanding the current in piecewise linear functions and testing with a similar

basis as opposed to piecewise constant expansion and testing functions would seem to be the

increase in convergence of the resulting series using the higher order bases. For the piecewise

constant expansion and testing functions, the Floquet series converges as O(p−4). However,

using piecewise linear expansion and testing functions, the convergence rate increases to

O(p−6). For even modest numbers of Floquet modes (Np=31), the solution has converged

to within 1% of the exact solution. Note the effect of the more complicated formulation

on the accuracy using the triangle bases is not immediately noticeable as a function of

increased Floquet contributions.

Table 2.2: Normalized TEx propagation constant kx0a for increasing number of unknownsfor f0=fa/c=1, εr=8.9, b=0.3545a, and Np=51

kx0a (pulse) kx0a (triangle)

Nx φ0=0 φ0=9.15 φ0=0 φ0=9.15

10 1.825 1.810 2.030 2.097

15 1.744 1.781 1.833 1.897

20 1.715 1.781 1.763 1.830

40 1.687 1.753 1.698 1.765

60 1.681 1.748 1.685 1.754

80 1.679 1.746 1.685 1.754

kx0a∗ 1.676 1.743 1.676 1.743

In Table 2.2, the normalized propagation constant for a normally incident plane wave

(φ0=0) and off-axis incident plane wave (φ0=9.15) for the TEx case is shown as func-

tion of the number of unknowns (Nx) and unknown basis set (pulse or triangle) for the

combination of normalized frequency f0=fa/c=1, 51 Floquet modes (Np), relative permit-

tivity εr=8.9, and filling fraction b/a=0.3545. Again, the results presented in Table 2.2

confirm that the necessary convergence occurs for both the pulse and triangle bases cases

for normal incidence and off-axis propagation as the number of unknowns increase. For

normalized frequency equal to unity (λ0=a), the electrical length of the dielectric insert

is 0.3545λ0/√εr=0.1188/λ0. Thus, only a few unknowns should be required to accurately

model the physical problem. For example, the solution obtained using only Nx=20 un-

knowns is within 5% of the exact solution. However, it is clear from the data that even

36

though the Floquet series may converge more quickly using higher order bases, the error

introduced to the final solution is dominated by the number of subsectional bases. Thus,

using the minimum number of bases if the most efficient solution to the problem.

The computational cost of computing each individual element in the impedance matrix

is proportional to Np. Although the implementation of pulse basis functions provides a

more slowly converging Floquet series than does the implementation using triangular bases,

it achieves a more accurate solution using fewer subsectional bases. The piecewise linear

functions are not only more expensive to compute than the simple pulse functions but are

also more prone to coding error. Since the total cost of computing the full impedance matrix

is proportional to N2xNp, incorporating fewer basis functions necessarily decreases this cost.

Tables 2.1 and 2.2 clearly show that the accuracy of computing the propagation constant is

more sensitive to the number of subsectional bases used in the approximation of the current

than the number of Floquet modes used to compute the Floquet series.

Table 2.3: Normalized TMx propagation constant kx0a for increasing number of Floquetmodes for f0=fa/c=1, εr=8.9, b=0.3545a, and Nx=40

kx0a (pulse) kx0a (triangle)

Np φ0=0 φ0=9.15 φ0=0 φ0=9.15

7 2.471 2.538 2.460 2.544

11 1.723 1.793 1.712 1.800

21 1.691 1.763 1.681 1.775

31 1.688 1.761 1.678 1.767

41 1.687 1.760 1.696 1.767

51 1.687 1.760 1.696 1.767

kx0a∗ 1.676 1.751 1.676 1.751

In Table 2.3, the normalized propagation constant for a normally incident plane wave

(φ0=0) and off-axis incident plane wave (φ0 = 9.15) for the TMx case is shown as func-

tion of the number of unknowns (Nx) and unknown basis set (pulse or triangle) for the

combination of normalized frequency f0=fa/c=1, 51 Floquet modes (Np), relative permit-

tivity εr=8.9, and filling fraction b/a=0.3545. As in the TEx case, the value φ0=9.15

corresponds to ky0=1/a, the inverse of the unit cell size. In Table 2.4, the normalized prop-

agation constant for a normally incident plane wave (φ0=0) and off-axis incident plane

wave (φ0=9.15) for the TEx case is shown as function of the number of Floquet modes

37

(Np) and unknown basis set (pulse or linear) for the combination of normalized frequency

f0=fa/c=1, 40 subsectional unknowns (Nx), relative permittivity εr=8.9, and filling frac-

tion b/a=0.3545.

Table 2.4: Normalized TMx propagation constant kx0a for increasing number of unknownsfor f0=fa/c=1, εr=8.9, b=0.3545a, and Np=51

kx0a (pulse) kx0a (triangle)

Nx φ0=0 φ0=9.15 φ0=0 φ0=9.15

10 1.824 1.898 2.029 2.093

15 1.744 1.817 1.834 1.901

20 1.715 1.788 1.765 1.833

40 1.687 1.760 1.696 1.767

60 1.681 1.755 1.685 1.760

80 1.679 1.753 1.685 1.751

kx0a∗ 1.676 1.751 1.676 1.751

Similar conclusions are drawn in Tables 2.3 and 2.4 for the TMx case that are ob-

served for the TEx case. The accuracy increases both by increasing the number of Floquet

mode contributions and by modeling the equivalent current more carefully by increasing the

number of unknowns. A simple pulse basis expansion serves to solve the problem quickly

and accurately. This observation will hold true as well for the two-dimensional structures

examined in Chapter 3.

In order to verify that the higher frequency solution of the propagation constants com-

puted using the method of moments technique and the plane wave expansion method are

correct, the band structure of a periodic array of dielectric slabs is calculated for a filling

fraction b/a=0.3545, relative dielectric constant εr=8.9, and off-axis4 propagation constant

ky0=1/a. Figure 2.9 shows the calculated band structure for both the TEx and TMx cases.

The moment method solution includes 40 subsectional bases and 51 Floquet modes. The

plane wave expansion method uses 63 Floquet modes. It is clear that the solutions ob-

tained by the two methods are in excellent agreement. For an off-axis propagation constant

of ky0 = 1/a and for relatively lower normalized frequency (f0 ≤ 0.4), the two modal so-

lutions (TEx, TMx) have different band gaps. At higher frequencies, the TEx and TMx

4By virtue of the fact that the TEx and TMx modes for are degenerate for normal incidence, an off-axispropagation case must be computed to be sure that the derivatives for the TMx case are formulated andimplemented correctly.

38

Normalized propagation constant (k xa)

0.00 0.79 1.57 2.36 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 TM (FS)TE (FS)TM (MoM)TE (MoM)

Figure 2.9: Band structure (TEx, TMx) of a periodic array of dielectric slabs for b/a=0.3545, εr=8.9, and off-axis propagation constant ky0=1/a computed usingmethod of moments technique (MoM) with Np=51 and plane wave expansion(PWE) solution with Np=63

bands are indistinguishable for the prescribed parameter set.

2.5 Experimental Verification

Experimental measurements with micromachined dielectrics were performed to verify

that a similar band structure predicted theoretically using the method of moment solution or

plane wave expansion (eigenvalue) solution for plane wave excitation of periodic media exists

for microstrip excitation of a finite height periodic dielectric grating. Experimental and

computational evidence indicates that the high dielectric constant material used to support

the microstrip circuit effectively confines the field to near the region of the microstrip itself.

Consequently, the quasi-TEM microstrip excitation simply “sees” a periodically changing

dielectric constant – much like the periodic array of dielectric slabs excited using plane

wave excitation. To corroborate the frequency response obtained through measurements of

fabricated circuits, finite element simulations using the Ansoft High-Frequency Structure

Simulator (HFSS) were carried out to model the exact structure of interest. Additionally, an

equivalent model for a microstrip mounted over a periodic dielectric substrate (hi-Z, low-Z

filter) was simulated using the Agilent EEsof EDA Series IV microstrip Libra component.

A one-dimensional periodic dielectric substrate, or grating, shown in Figure 2.10 was

39

a

b

t

Figure 2.10: Microstrip mounted on 1-D periodic dielectric substrate with period a, fillingfraction b/a=0.5, and height t

fabricated with period a=6.35 mm (250 mil) and filling fraction equal to 0.5 using Rogers

Corporation RT/duroid having a dielectric constant of 10.2 and a thickness t of 0.635 mm

(25 mil). The measured response of the milled one-dimensional grating microstrip circuit is

shown in Figure 2.11. It is observed that the first stopband, defined by the 10 dB bandwidth

in the figure, is centered around 12 GHz with a second gap located between 21 and 26 GHz.

A third gap may or may not be visible around 36 GHz.

Frequency (GHz)

0 10 20 30 40

Inse

rtio

n L

oss

(dB

)

-50

-40

-30

-20

-10

0

Measured (1-D grating)Measured (2-D lattice)

Figure 2.11: Measured frequency response of a microstrip mounted on a 1-D grating and2-D lattice substrate

Concurrent to the circuit fabrication, the band structure for a one-dimensional periodic

array of dielectric slabs with the same geometrical and electrical parameters (a=6.35 mm,

40

b=3.175 mm, and εr=10.2) was determined. Three stopgaps were found to exist within the

measurement range of the HP8510 Network Analyzer, the first between 8.0 and 13.2 GHz,

a second between 18.7 and 26.2 GHz, and another between 31.2 and 37.3 GHz. These

results are in agreement with the measured response shown in Figure 2.11. The simulated

response of an HFSS simulation of the structure and the response of a Libra simulation of an

equivalent model are plotted in Figure 2.12. Due to the computational cost of implementing

the HFSS simulation of the realizable structure, only frequencies between 5 and 15 GHz

Frequency (GHz)

0 10 20 30 40

Inse

rtio

n L

oss

(dB

)

-30

-20

-10

0

HFSS simulationLibra model

Figure 2.12: Simulated frequency response of HFSS and Libra simulations of a microstripmounted on a 1-D grating

were analyzed. Note that both simulations reveal the first gap around 12 GHz and the

Libra model predicts the second and third gaps at 24 GHz and 36 GHz, respectively. The

gaps determined from the measurements and simulations of the finite structure reveal one

noteworthy difference between the gaps (stopbands) produced by a finite lattice (filter) and

the true gap (stopband) produced by an infinite lattice (filter), namely, that using a small

number of lattice periods (five) necessarily changes the shape and center frequency of the

response of the “filter.”

The encouraging results led to the question of how much material could be removed

while maintaining the response of the system, ultimately shedding light on the question of

how well the quasi-TEMmicrostrip mode in these periodic circuits can be modeled by simple

arrays of planar dielectric slabs. To this end, three substrates were designed that successively

added material to the one-dimensional grating to produce a finite one-dimensional grating,

a two-dimensional lattice of square air rods (holes), and finally, a two-dimensional lattice

41

of circular holes. Incorporating circular rods in the design instead of square ones has the

advantage of increased ease and speed of machining the substrates using simple milling

procedures. Each of the new substrates shown in Figure 2.13 was designed to have a period

equal to that of the one-dimensional periodic substrate shown in Figure 2.10. Consequently,

(a)

(b)

(c)

(d)

Figure 2.13: Microstrip mounted on (a) 1-D grating, (b) finite 1-D grating, (c) 2-D latticeof square holes, and (d) 2-D lattice of circular holes

the microstrip line would traverse effectively similar sections of dielectric material.

HFSS simulations were carried out for each of the three models for frequencies between

5 and 15 GHz to see the effect of the added material on the bandwidth and location of

the first stopgap. The results of the finite element simulations of the three “equivalent”

circuits are shown in Figure 2.14. Also shown is the response of a through line mounted

on a homogeneous substrate of same thickness. Note that, for all intents and purposes, the

responses of all of the periodic gratings are the same.

Finally, a two-dimensional periodic dielectric substrate, shown in Figure 2.15, was fabri-

cated with period a=6.35 mm (250 mil), cell-to-cell distance c=a/√2=4.49 mm (177 mil),

and circular hole diameter b=3.17 mm (125 mil) using RT/duroid having a dielectric con-

stant of 10.2 and a thickness t of 0.635 mm (25 mil). The physical dimensions of the two-

dimensional lattice were designed to produce the exact period a used in the one-dimensional

periodic substrate. The band structure was originally adapted from band plots presented

in [63] but has since been verified by solutions derived in Chapter 3. The frequency response

of the two-dimensional periodic substrate is also included in Figure 2.11.

42

Frequency (GHz)

5.0 7.5 10.0 12.5 15.0

Inse

rtio

n L

oss

(dB

)

-30

-25

-20

-15

-10

-5

0

Through line1-D gratingFinite 1-D grating2-D lattice of square holes2-D lattice of circular holes

Figure 2.14: Simulated frequency response of a microstrip mounted on a 1-D grating, finite1-D grating, 2-D lattice of square holes, and 2-D lattice of circular holes

a

t

d

c

Figure 2.15: Microstrip mounted on two-dimensional periodic dielectric substrate with pe-riod a, cell-to-cell distance c=a/

√2, hole diameter d, and height t

Noticeable differences in the frequency response of the one-dimensional grating and two-

dimensional lattice are observed in Figure 2.11. Not only are the bandwidths of the first two

stopbands of the two-dimensional substrate lattice narrower than the ones produced by the

one-dimensional grating, the center frequencies are shifted slightly. The HFSS simulations

shown in Figure 2.14 would seem to indicate that for microstrip excitation, the responses of

the one- and two-dimensional lattices shown in Figures 2.10 and 2.15 are indistinguishable.

This observation raises the question of how well HFSS can model the finite periodic dielectric

substrate.

43

2.6 Conclusions

The solution for the propagation of electromagnetic energy through a one-dimensional

periodic dielectric structure has been determined using two distinct solution techniques. A

Fourier series solution has been implemented by solving the one-dimensional (differential)

wave equation for periodic functions. Concurrently, a separate solution is obtained by de-

riving the integral equation solution for a one-dimensionally periodic media. The advantage

of using the plane wave expansion method over the IE/MoM solution is the computational

speed. The optimization procedure used in the IE/MoM solution requires the impedance

matrix to be filled for each iteration of the computation of the propagation constant. Even

using a highly efficient optimizer to quickly find the eigenvalue of interest, the process can

be somewhat time-consuming. For one-dimensional structures, this does not present a se-

rious problem. However, it will be clearly shown that for the two dimensional lattices of

Chapter 3, this restriction has grave consequences. The accuracy of the plane wave expan-

sion method increases with increased Floquet mode contributions whereas the accuracy of

the IE/MoM solution increases both by increasing the number of Floquet mode contribu-

tions and by modeling the equivalent current more carefully by increasing the number of

unknowns. It is observed that, for the IE/MoM solution, a simple pulse basis expansion

serves to solve the problem quickly and accurately. This observation will hold true as well

for the two-dimensional structures examined in Chapter 3.

Experimental verification of the one-dimensional band gaps is accomplished by determin-

ing the filtering effect of mounting a microstrip over a one-dimensional periodic substrate.

The quasi-TEM fields of the microstrip excitation for the periodic substrate are modeled as

plane waves incident on a one-dimensionally periodic medium. The band structure of the

resulting microwave circuit is also modeled effectively using a simple hi-Z, low-Z filter and

is similar to the band structure determined from the plane wave solution.

44

CHAPTER 3

Two-Dimensional Periodic Dielectric Structures

3.1 Introduction

As opposed to one-dimensional structures, where many applications are found at the

higher quasi-optical and optical frequencies [1, 2], research into applications involving two-

dimensional periodic structures is fruitful at microwave and lower frequencies. In fact, an

entire class of artificial surfaces and/or materials termed frequency selective surfaces (FSS)

incorporates the use of period to provide unique spectral characteristics in applications rang-

ing from radar cross section reduction to antenna array beam-forming. Two-dimensional

dielectric lattices are actively being incorporated into microwave circuit and antenna de-

signs, high-Q filters and resonators, waveguides, and novel materials/surfaces. Although

serious attention has been focused by the electromagnetics community on traditional fre-

quency selective surfaces, new applications have recently emerged for more general two-

dimensional periodic structures and are challenging traditional concepts and views about

electromagnetic propagation in periodic media [3, 4, 5].

This chapter extends the solutions and techniques developed for one-dimensional period

in Chapter 2 to structures that have periodicities in two directions. The phase constants for

electromagnetic waves propagating in a two-dimensional lattice for both dielectric rods in

an air background and air columns immersed in a dielectric media are found explicitly using

two similar but decidedly different techniques. The first solution, detailed in Section 3.2,

incorporates the use of a Fourier series representation for the periodic field and is determined

by solving the differential equation (two-dimensional wave equation) for periodic media.

The second solution is found by deriving an integral equation for the periodic field and

numerically solving the resulting linear system and is presented in Section 3.3.

45

In light of the complicated nature of two-dimensional periodic media, effective medium

theory (EMT) is applied to the lattice which reduces the problem to that of solving for

the propagation in an equivalent one-dimensional array with an effective permittivity. In

Section 3.4, the theory of EMT is outlined briefly and its advantages and limitations are

discussed. One of the significant disadvantages of EMT is its limitation to electrically small

lattice elements. A number of representative structures are used in the chapter to show the

behavior of electromagnetic fields in two-dimensional periodic media and observations about

the salient features of each are discussed. The particular application of a two-dimensional

periodic dielectric structure in parallel-plate mode reduction in conductor-backed slots is

presented extensively in Appendix D. For the structures shown in this chapter, only in-

plane propagation is considered. This is significant because complete band gaps1 only exist

for in-plane propagation.

1A complete band gap (for a given polarization) is defined as one where over a range of frequencies(band), electromagnetic energy may not propagate in any direction. Some authors define a complete bandgap as one where for any electromagnetic wave polarization, energy may not propagate in any direction.

46

3.2 Plane Wave Expansion Method

3.2.1 Analytical Techniques

For two-dimensional periodic problems, the solution of the exact eigenvalue equation can

be obtained through the use of a double Fourier series. A representative two-dimensional

E(H)

φ

ba

y

xz

Figure 3.1: Cross-sectional view of two-dimensional array of dielectric rods of diameter b ina periodic square lattice with period a

periodic array of dielectric rods of diameter b in a periodic square lattice2 with period a is

illustrated in cross section in Figure 3.1. If the electric field has only an axial (or z -directed)

component, the mode is transverse magnetic to z and is denoted TMz. If the electric field

has only components in the transverse x-y plane, the mode is transverse electric to z and

is denoted TEz.

TMz Case

The electric field is expanded as a periodic function with period a in the x -direction and

prescribed propagation constant kx0 and with period c in the y-direction and prescribed

propagation constant ky0 and is given by

E(x, y) = zEz(x, y) = zEp(x, y) e−jkx0x e−jky0y (3.1)

2Square lattices are used in this chapter to simplify the notation and understanding. Extensions of thesolutions to other two-dimensional lattices depicted in Figure B.3 of Appendix B is straightforward.

47

where Ep(x, y) is the periodic electric field that propagates only in the xy-plane. Again,

applying the operator (∇2xy + k2) to the electric field in (3.1) yields an equation similar

to (2.15) with εr(x) replaced with εr(x, y). If the rods are assumed to be infinite in the z

direction(∂∂z = 0

), then the wave equation simplifies to

− ∂2

∂x2Ez(x, y)− ∂2

∂y2Ez(x, y) = k2

0εr(x, y)Ez(x, y) (3.2)

If the periodic field, expanded as in a double Fourier series in x and y with unknown

coefficients anq which serve to represent the dependence on z,

Ep(x, y) =∑n

∑q

anq e−j 2πn

ax e−j

2πqcy, (3.3)

and the periodic dielectric rods, expanded in another double Fourier series with coefficients

bmp,

εr(x, y) =∑m

∑p

bmp e−j 2πm

ax e−j

2πpay, (3.4)

are substituted into (3.2) and the algebraic operations are carried out, then

∑n

∑q

[(2πna

+ kx0

)2

+(2πqc

+ ky0

)2]anq e

−j 2πnax e−j

2πqcy =

k20

∑m

∑p

bmp e−j 2πm

ax e−j

2πpay∑n

∑q

anq e−j 2πn

ax e−j

2πqcy. (3.5)

In order to determine the coefficients anq and bmp, (3.5) is multiplied by orthogonal functions

in x and y and integrated over the unit cell. This integration produces a Kronecker delta

function for specific indices in the x and y directions and

∑n

∑q

[(2πna

+ kx0

)2

+(2πqc

+ ky0

)2]anqδ

(2πlc

− 2πqc

)δ

(2πka

− 2πna

)=

k20

∑m

∑p

∑n

∑q

anqbmpδ

(2πlc

− 2πpc

− 2πqc

)δ

(2πka

− 2πma

− 2πpa

). (3.6)

The convolution in (3.6) can be cast, albeit not as easily as (2.20), into the following general

matrix form

anq

[(2πna

+ kx0

)2

+(2πqc

+ ky0

)2]= k2

0

∑m

∑p

anqbn−m,q−p (3.7)

where for the square dielectric rod illustrated in Figure 3.2(a),

48

a

d

b

c

(a)

a

b

(b)

Figure 3.2: Unit cell dimensions for (a) rectangular dielectric rods and (b) circular dielectricrods where for the rectangular rod, a and c are the unit cell sizes, b and d arethe element edge lengths, and for the circular rod, a is the unit cell size and bis the diameter

bn−m,q−p =1ac

b/2∫−b/2

d/2∫−d/2

(εr − 1) e−j2π(n−m)

ax e−j

2π(q−p)a

y dx dy

+1ac

a/2∫−a/2

c/2∫−c/2

(1) e−j2π(n−m)

ax e−j

2π(q−p)a

y dx dy

=bd

ac(εr − 1) sinc

π(n−m)ba

sincπ(q − p)d

c+ δn−m,q−p (3.8)

and for the circular dielectric rod illustrated in Figure 3.2(b),

bn−m,q−p =2πa2

b/2∫0

(εr − 1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

+2πa2

a∫0

(1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

=πb2

a2(εr − 1)

2J1

(√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2)

√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2+ δn−m,q−p. (3.9)

In Figure 3.2(a), a and c are the unit cell sizes, b and d are the square element edge lengths.

In Figure 3.2(b), a is the unit cell size and b is the diameter of the circular element. With

a little algebraic work, the coefficients bn−m,q−p in (3.8) and (3.9) can be manipulated to

agree with similar expressions in [107] and [73].

49

If the dielectric rods are replaced by air columns, the coefficients bn−m,q−p become

bn−m,q−p =1ac

a/2∫−a/2

c/2∫−c/2

(εr) e−j2π(n−m)

ax e−j

2π(q−p)a

y dx dy

− 1ac

b/2∫−b/2

d/2∫−d/2

(εr − 1) e−j2π(n−m)

ax e−j

2π(q−p)a

y dx dy

=εrδn−m,q−p − (εr − 1)bd

acsinc

π(n−m)ba

sincπ(q − p)d

c(3.10)

for the rectangular air column shown in Figure 3.3(a), and

a

d

b

c

(a)

a

b

(b)

Figure 3.3: Unit cell dimensions for (a) rectangular dielectric air columns and (b) circularair columns where for the rectangular column, a and c are the unit cell sizes, band d are the element edge lengths, and for the circular column, a is the unitcell size and b is the diameter

bn−m,q−p =2πa2

a∫0

(εr)J0

√(

2π(n−m)ra

)2

+(2π(q − p)r

a

)2 r dr

− 2πa2

b/2∫0

(εr − 1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

=εrδn−m,q−p − (εr − 1)πb2

a2

2J1

(√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2)

√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2(3.11)

for the circular air column shown in Figure 3.3(b).

A generalized linear eigensystem problem is represented by Ax = λBx where A and B

are n×n matrices. The value λ is an eigenvalue and x = 0 is the corresponding eigenvector.

50

The propagating modes in the TMz case are solutions of the generalized linear eigensystem

in (3.7).

TEz Case

For the TEz case, the electric field is replaced by the magnetic field in the wave equation

∇×

1εr(x, y)

∇× zHz(x, y)+ k2

0zHz(x, y) = 0 (3.12)

where εr(x) in (2.23) has been replaced with εr(x, y). The same care must be taken to

apply the wave equation operator to the magnetic field in the two dimensional case that

was taken to apply it to the magnetic field in the one dimensional case. The resulting

equation resembles (2.24)

1εr(x, y)

z · ∇ ×∇× H(x, y) + z · ∇

1εr(x, y)

×∇× H(x, y) = −k2

0Hz(x, y). (3.13)

Expanding the periodic magnetic field in a double Fourier series,

Hp(x) =∑n

∑q

anq e−j 2πn

ax e−j

2πqcy, (3.14)

and the inverse of the dielectric function in another double Fourier series as

1εr(x, y)

=∑m

∑p

bmp e−j 2πm

ax e−j

2πpay, (3.15)

substituting the expansions into (3.13), and carrying out the curl and gradient operations

yields

∑m

∑p

bmp e−j 2πm

ax e−j

2πpay

− ∂2

∂x2

∑n

∑q

anq e−j 2πn

ax e−j

2πqcy e−jkx0xe−jky0y

− ∂2

∂y2

∑n

∑q

anq e−j 2πn

ax e−j

2πqcy e−jkx0x e−jky0y

− ∂

∂x

∑m

∑p

bmp e−j 2πm

ax e−j

2πpay ∂

∂x

∑n

∑q

anq e−j 2πn

ax e−j

2πqcy e−jkx0x e−jky0y

− ∂

∂y

∑m

∑p

bmp e−j 2πm

ax e−j

2πpay ∂

∂y

∑n

∑q

anq e−j 2πn

ax e−j

2πqcy e−jkx0x e−jky0y

= −k20

∑n

∑q

anq e−j 2πn

ax e−j

2πqcy e−jkx0x e−jky0y. (3.16)

Simplifying (3.16) by carrying out the derivatives and combining like series, integrating the

result over one period, and simplifying the resulting expression yields an equation similar

to (2.28)

51

∑m

∑p

anqbn−m,q−p

[(2πna

+ kx0

)2

+(2πqc

+ ky0

)2

− 2π(n−m)a

(2πna

+ kx0

)− 2π(q − p)

c

(2πqc

+ ky0

)]= −k2

0anq (3.17)

where for rectangular rods,

bn−m,q−p =1ac

b/2∫−b/2

d/2∫−d/2

(1εr

− 1)e−j

2π(n−m)a

x e−j2π(q−p)

ay dx dy

+1ac

a/2∫−a/2

c/2∫−c/2

(1)e−j2π(n−m)

ax e−j

2π(q−p)a

y dx dy

=bd

ac

(1εr

− 1)sinc

π(n−m)ba

sincπ(q − p)d

c+ δn−m,q−p (3.18)

and for circular rods,

bn−m,q−p =2πa2

b∫0

(1εr

− 1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

+2πa2

a∫0

(1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

=πb2

a2

(1εr

− 1) 2J1

(√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2)

√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2+ δn−m,q−p (3.19)

Again if the dielectric rods are replaced by air columns as before, the coefficients bn−m,q−p

become

bn−m,q−p =1ac

a/2∫−a/2

c/2∫−c/2

(1εr

)e−j

2π(n−m)a

x e−j2π(q−p)

ay dx dy

− 1ac

b/2∫−b/2

d/2∫−d/2

(1εr

− 1)e−j

2π(n−m)a

x e−j2π(q−p)

ay dx dy

=1εrδn−m,q−p −

(1εr

− 1)bd

acsinc

π(n−m)ba

sincπ(q − p)d

c, (3.20)

52

for the rectangular air column shown in Figure 3.3(a), and

bn−m,q−p =2πa2

a∫0

(1εr

)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

− 2πa2

b/2∫0

(1εr

− 1)J0

√(

2π(n −m)ra

)2

+(2π(q − p)r

a

)2 r dr

=1εrδn−m,q−p −

(1εr

− 1)πb2

a2

2J1

(√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2)

√(2π(n−m)b

a

)2+(

2π(q−p)ba

)2, (3.21)

for the circular air column in Figure 3.3(b). The explicit expressions for the coefficients

bn−m,q−p in (3.10), (3.11), (3.18), (3.19), (3.20), and (3.21) have not been found in the

literature.

An ordinary linear eigensystem problem is represented by the equation Ax = λx where

A denotes an n × n matrix. The propagating modes in the TEx case are solutions of the

ordinary eigensystem problem in (3.17).

3.2.2 Matrix Solution of the Eigensystem

The resulting eigenvalues of the matrices in (3.7) and (3.17) are the squares of the

frequencies of the propagating modes in the structure. The solutions of the frequencies

of the propagating modes in the structure are found for specific values of kx0a, ky0a ∈[0, 2π]. The full band structure3 (TMx) for a normally incident mode in a periodic array of

dielectric rods with b/a=0.3545 and εr=8.9 is shown in Figure 3.4. The values for the filling

fraction and the relative dielectric constant are obtained from a two-dimensional structure

in [64, 129]. The Brillouin zone for a square lattice is also included in the figure where

the Γ, X, and M points are defined in Appendix B. A complete TM band gap exists over

a significant range of frequencies. Certainly, larger incomplete bands exist (particularly

in the Γ–M and Γ–X directions) where propagation is allowed for specific directions and

these gaps are effectively used in designs that do not need total omni-directional stopbands.

Notice that there exists no complete TE band gap. In fact, only a small TE gap exists in

any direction (note the small gap at the X point in Figure 3.4) for this geometrical and

electrical configuration.3A full band structure is one where the propagation constant is determined along the edges of the Brillouin

zone. Theoretically, one must determine the propagation constant for all of the unique combinations of kx0

and ky0 . Fortunately, physics dictates that the band gaps are found at the edges of the Brillouin zone (BZ).Thus, by sampling the edge of the BZ, one can quickly determine the requisite full band structure.

53

0.00 3.14 6.28 9.42

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

0.75

1.00

TMz

TEz modes

modes

band gapTM

ΓX

M

Γ βx (βy=0) X βy (βx=π/a) M βx=βy Γ

Normalized propagation constant ( βa)

Figure 3.4: Band structure (TMz) for a normally incident mode in a periodic array ofdielectric rods with b/a=0.3545 and εr=8.9

Extensive research has concluded that large complete TM band gaps are most likely to be

found using isolated regions of high dielectric constant material imbedded in a background of

lower dielectric constant, whereas for TE band gaps, significant bands are found for isolated

regions of lower dielectric constant immersed in a higher dielectric background [38, 129].

The latter conclusion can be seen in Figure 3.5 where the full band structure (TEx) for a

normally incident mode is shown for the periodic array of air columns with b=0.6455a and

background dielectric constant εr=8.9. For this structure, which is complementary to the

structure in Figure 3.4, no complete TM gap can be found. Small TM gaps been seen at

the X and M points in the lattice but are too small to be of much value.

If the structures are viewed from a connectivity point of view, new insights can be

gleaned from the preceding figures. The air cross of Figure 3.6(a) is equivalent to the

dielectric square of Figure 3.2(a). Thus, regions of air that are “connected” to other cells

by thin veins of air are useful for TM band gaps. The complementary dielectric cross of

Figure 3.6(b) is equivalent to the air square of Figure 3.3(a). For useful TE band gaps,

regions of dielectric should be connected to other cells by thin veins of dielectric. Either

interpretation reveals that the electric field maintains a preference to distribute itself in

certain ways depending on the polarization.

A simple check of the correctness of the results obtained from the two-dimensional pe-

riodic structure is to allow the filling fraction in one lattice direction to increase until an

54

0.00 3.14 6.28 9.42

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

TMz modes

TEz modes

Γ βx (βy=0) X βy (βx=π/a) M βx=βy Γ

Normalized propagation constant ( βa)

band gapTE

ΓX

M

Figure 3.5: Band structure (TEz) for a normally incident mode in a periodic array of aircolumns with b/a=0.6455 and εr=8.9

(a) (b)

Figure 3.6: Equivalent structures for Figure 3.2(a) and Figure 3.3(a)

equivalent one-dimensionally periodic structure is obtained. As the unit cell geometry ap-

proaches the one-dimensional periodic structure of equal filling fraction, the propagation

constant should approach the value of the one-dimensional propagation constant (ky=0). In

Figure 3.7 the normalized TMz propagation constant kx0a is shown as a function of d/c for

normalized frequency f0=fa/c=0.5, relative dielectric constant εr=10.2, normal incidence,

and filling fraction b/a=0.2. Comparing the results with the normalized propagation con-

stant obtained from the equivalent one-dimensional structure, k1Dx0a, one can conclude that

indeed the solutions are the same. Similar results can be obtained for the normalized TEz

propagation constant kx0a as a function of d/c.

55

k =2.85 k =2.58 k =2.33k =2.43 k =2.210k =2.24x xxxxx

Ez

x

y

d/c=0.2 d/c=0.95d/c=0.8d/c=0.6d/c=0.4 d/c=1.0

b/a=0.2

k =2.204x1D

Figure 3.7: Normalized TMz propagation constant kx0a as a function of d/c for f0=fa/c=0.5, εr=10.2, ky=0, and b/a=0.2

3.3 Formulation of Integral Equations

Great physical insight is often obtained through the necessary theoretical formulation

of a problem from basic principles. This is particularly true when deriving two- and three-

dimensional integral equations (IE) that when solved, yield the unknowns of interest. Ap-

plying the method of moments (MoM) numerical solution technique, where the physics of

the problem is often intimately related to the formulation itself, also provides intuition

about the quantities being studied.

3.3.1 Derivation of Electric Field Integral Equation

Similarly to the derivation of the one-dimensional case, the dielectric rods are replaced

by equivalent (polarization) volume currents and the total field is determined as the sum

of the incident field produced by a known source with the dielectric absent and a scattered

field contributed by the equivalent currents induced in the periodic scatterers.

TMz Case

Consider the representative dielectric rod (period not shown) of diameter b in a periodic

square lattice with period a in Figure 3.8. The rods are excited by an incident time harmonic

(ejωt) plane wave propagating in a direction normal to the z axis (in-plane propagation)

with its electric field parallel to the axis (TMz) which induces an electric current J = Jz z

with a z component only. Note that, for uniformity in the z direction, ∇ · J of (2.31)

is zero. The total electric field everywhere is computed as the sum of the scattered field

produced by the equivalent induced electric current and the incident electric field given

by Ei(x, y) = zE0e−jk0(x cosφ0+y sinφ0), which impinges upon the rod along a ray in the

56

E(H)

y

xz

φ

b

a

Figure 3.8: Representative dielectric rod of diameter b in a periodic square lattice withperiod a

k = (x cosφ0+y sinφ0) direction defined by an angle φ0 with respect to the x axis (φ0 = 0).

The first step is to replace the dielectric material with equivalent volume currents using

(2.37). If we assume for simplicity that the two-dimensional periodic array in Figure 3.1

is composed of dielectric rectangular rods of cross section b×d in a periodic square lattice

with cross section a×c, then for the TMz case,

Jz(x, y) = jk0Y0 (εr(x, y)− 1)Ez(x, y) (3.22)

where εr(x, y) is the permittivity function of the rods defined by

εr(x, y) =

εr, −b/2 < x < b/2,−d/2 < y < d/2;1, otherwise.

(3.23)

Since the dielectric material is periodic in x with period a and in y with period c, the

resulting equivalent currents must satisfy

Jz(x+ pa, y + qc) = Jz(x, y)e−jkx0pae−jky0qc (3.24)

for phase shifts kx0a in the x direction and ky0c in the y direction. Using (2.31), the

scattered field is determined from the periodic equivalent currents (3.24) to be

Esz(x, y) = −jk0Z0

b2∫

− b2

d2∫

− d2

Jz(x′, y′)Gp(x, y;x′, y′) dx′ dy′ (3.25)

57

where Gp(x, y;x′, y′) is the two-dimensional periodic free-space Green’s function (PFSGF)

given by

Gp(x, y;x′, y′) =∑p,q

e−jkx0pa e−jky0qc14jH

(2)0

(k0

√(x− x′ − pa)2 + (y − y′ − qc)2

)

(3.26)

and p, q are the Floquet mode indices for propagation in the x and y directions, re-

spectively. Using Ez(x, y) = Esz(x, y) + Ei

z(x, y), one can formulate the following EFIE to

determine the equivalent currents

jk0Z0

b2∫

− b2

d2∫

− d2

Jz(x′, y′)Gp(x, y;x′, y′) dx′ dy′ +Jz(x, y)

jk0Y0 (εr(x, y)− 1)= Ei

z(x, y). (3.27)

Two-dimensional periodic free-space Green’s function The extension of the one-

dimensional periodic free-space Green’s function to a periodic two-dimensional periodic

free-space Green’s function is straight forward. The spatial form of the two-dimensional

free-space periodic Green’s function can also be easily transformed into its spectral form

through the use of the Poisson sum formula [70]. From Table I of [51],

∑p,q

f(p, q) =∑p,q

14jH

(2)0

(k

√(x− pa)2 + (y − qc)2

)

=1a

∑p,q

1

2j

√k2 −

(2πpa

)2e−j

√k2−( 2πp

a )2|y−qc|e−j2πp

ax.

(3.28)

Again, using Table I of [51],

1a

∑p,q

1

2j

√k2 −

(2πpa

)2e−j

√k2−( 2πp

a )2|y−qc|e−j2πpax

= − 1ac

∑p

1[k2 −

(2πpa

)2 −(

2πqc

)2]e−j 2πp

axe−j

2πqcy (3.29)

where a is the unit cell width in the x-direction, b is the width of the dielectric region in

the x-direction, c is the unit cell depth in the y-direction, d is the depth of the dielectric

region in the y-direction, k is the wavenumber of the medium, and p, q are the Floquet

indices. Equation (3.29) becomes upon substitution of a phase shift k = kx0 x+ ky0 y

∑p,q

14jH

(2)0

(k

√(x− pa)2 + (y − qc)2

)e−jkx0pae−jky0qc = − 1

ac

∑p,q

e−jkxpxe−jkyqy

k2 − k2xp

− k2yq

(3.30)

58

where

kxp =2πpa

+ kx0

kyq =2πqc

+ ky0 .

Thus, (3.27) becomes upon substitution

− jk0Z0

ac

b2∫

− b2

d2∫

− d2

Jz(x′, y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′ +Jz(x, y)

jk0Y0 (εr(x, y)− 1)

= Eiz(x, y). (3.31)

TEz Case

Similarly for the the TEz case, we replace the dielectric material with equivalent vol-

ume currents. However, note that since the rods are excited by an incident plane wave

propagating in a direction normal to the z axis (in-plane propagation) with its magnetic

field parallel to the axis (TEz), the induced electric current J has only components in the

transverse (x,y) plane.

In this case, the dielectric material is replaced with equivalent volume currents for both

the x and y components

Jx(x, y) = jk0Y0 (εr(x, y)− 1)Ex(x, y) (3.32)

Jy(x, y) = jk0Y0 (εr(x, y)− 1)Ey(x, y). (3.33)

Substituting the periodic polarization currents into (2.31), the scattered field can be deter-

mined to be

Esx(x, y) = −jk0Z0

(1 +

1k20

∂2

∂x2

) b2∫

− b2

d2∫

− d2

Jx(x′, y′) Gp(x, y;x′, y′) dx′ dy′

− Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jy(x′, y′) Gp(x, y;x′, y′) dx′ dy′ (3.34a)

59

Esy(x, y) = − Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jx(x′, y′) Gp(x, y;x′, y′) dx′ dy′

− jk0Z0

(1 +

1k20

∂2

∂y2

) b2∫

− b2

d2∫

− d2

Jy(x′, y′) Gp(x, y;x′, y′) dx′ dy′. (3.34b)

Using (2.32), one can formulate the following coupled EFIEs to determine the equivalent

currents

jk0Z0

(1 +

1k20

∂2

∂x2

) b2∫

− b2

d2∫

− d2

Jx(x′, y′) Gp(x, y;x′, y′) dx′ dy′

+Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jy(x′, y′) Gp(x, y;x′, y′) dx′ dy′ +Jx(x, y)

jk0Y0 (εr(x, y)− 1)= Ei

x(x, y)

(3.35a)

Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jx(x′, y′) Gp(x, y;x′, y′) dx′ dy′ +Jy(x, y)

jk0Y0 (εr(x, y) − 1)

+ jk0Z0

(1 +

1k20

∂2

∂y2

) b2∫

− b2

d2∫

− d2

Jy(x′, y′) Gp(x, y;x′, y′) dx′ dy′ = Eiy(x, y). (3.35b)

Using the Poisson sum formula (3.30) shown on page 58, (3.35) becomes

jk0Z0

(1 +

1k20

∂2

∂x2

) b2∫

− b2

d2∫

− d2

Jx(x′, y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

+Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jy(x′, y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′ +Jx(x, y)

jk0Y0 (εr(x, y) − 1)

= Eix(x, y) (3.36a)

60

Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Jx(x′, y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′ +Jy(x, y)

jk0Y0 (εr(x, y)− 1)

+ jk0Z0

(1 +

1k20

∂2

∂y2

) b2∫

− b2

d2∫

− d2

Jy(x′, y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

= Eiy(x, y). (3.36b)

3.3.2 Method of Moments Formulation

For two-dimensional problems, the choice of discretization can have a significant effect

on the reduction of the integral equations into matrix equations. After discretizing the

geometry and expanding the unknown equivalent electric current across the dielectric region

with a linear combination of subsectional or entire domain basis functions, the equations

are tested in order to obtain an adequate number of equations to solve for the unknown

coefficients of the basis functions. In order to provide verification of the IE/MoM procedure,

computer codes have been written in Fortran to determine the propagating modes of a

periodic array of dielectric slabs.

TMz Case

Piecewise constant expansion / Piecewise constant testing For piecewise constant

functions, we discretize the dielectric region into N=NxNy subsections of width ∆x = b/Nx

and ∆y = d/Ny, respectively, to approximate the original surface and represent Jz by a

linear combination of N piecewise constant basis functions Πn(x, y) with unknown current

coefficients Jn

Jz(x, y) =N∑n=1

JnΠn(x, y) =N∑n=1

JnΠn(x)Πn(y) (3.37)

where

Πn(x) =

1, xn − ∆x

2 < x < xn + ∆x2 ;

0, otherwise;(3.38)

and

Πn(y) =

1, yn − ∆y

2 < y < yn +∆y

2 ;

0, otherwise;(3.39)

61

and xn = − b2 +∆x(n− 1

2) and yn = −d2 +∆y(n− 1

2).

Substituting the currents into (3.31) yields

N∑n=1

Jn

jk0Z0

ac

xn+∆x2∫

xn−∆x2

yn+∆y2∫

yn−∆y2

∑p,q

e−jkxp(x−x′)e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dy′ dx′ − Πn(x)Πn(y)jk0Y0 (εr(x, y)− 1)

= 0. (3.40)

Testing (3.40) with piecewise constant functions centered at (xm, ym) yields the following

equation

N∑n=1

Jn

jk0Z0

ac

xm+∆x2∫

xm−∆x2

ym+∆y2∫

ym−∆y2

xn+∆x2∫

xn−∆x2

yn+∆y2∫

yn−∆y2

∑p,q

e−jkxp(x−x′)e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dy′ dx′ dy dx

−xm+∆x

2∫xm−∆x

2

ym+∆y2∫

ym−∆y2

Πn(x)Πn(y)jk0Y0 (εr(x, y) − 1)

dx dy

= 0, m = 1, 2, . . . , N. (3.41)

By carrying out the integrations in (3.41) and simplifying the resulting expressions, the

impedance matrix elements can be determined explicitly and are listed in Appendix C. The

resulting matrix equation is written in a form similar to (2.54).

The transforms of the testing and expansion functions, denoted Πm(kx) and Πn(kx),

are given in (2.56). The transforms of the expansion and testing functions, denoted Πm(ky)

and Πn(ky), respectively, are

Πm(ky) = ∆ sinc(ky∆2

)e−jkyym (3.42)

Πn(ky) = ∆ sinc(ky∆2

)ejkyyn . (3.43)

Piecewise linear expansion / Piecewise linear testing For piecewise linear functions,

we discretize the dielectric region into N=NxNy subsections of width ∆x = b/(Nx− 1) and

∆y = d/(Ny−1), respectively, that approximate the original surface and approximate Jz by

a linear combination of N piecewise linear basis functions Λn(x, y) with unknown current

coefficients Jn

Jz(x, y) =N∑n=1

JnΛn(x, y) =N∑n=1

JnΛn(x)Λn(y) (3.44)

62

where

Λn(x) =

1− |x−xn|

∆x, xn −∆x < x < xn +∆x;

0, otherwise,(3.45)

and

Λn(y) =

1− |y−yn|

∆y, yn −∆y < y < yn +∆y;

0, otherwise,(3.46)

where xn = − b2 +∆x(n− 1) and yn = −d

2 +∆y(n− 1). Substituting (3.44) into (3.31) and

testing with piecewise linear basis functions yields

N∑n=1

Jn

jk0Z0

ac

xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′)e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dy′ dx′ dy dx

− Λn(x)Λn(y)jk0Y0 (εr(x, y)− 1)

= 0. (3.47)

The impedance matrix elements shown in Appendix C can be determined evaluating the

integrals in (3.47).

The transforms of the testing and expansion functions, denoted Λm(kx) and Λn(kx) are

given in (2.60). The transforms of the testing and expansion functions, denoted Λm(ky)

and Λn(ky) are

Λm(ky) = ∆ sinc2(ky∆2

)e−jkyym

Λn(ky) = ∆ sinc2(ky∆2

)ejkyyn .

The transforms of the two “half-basis” functions, denoted Λ1(kx) and ΛN (kx), are given in

(2.61). The transforms of the two “half-basis” functions, denoted Λ1(ky) and ΛN (ky), are

Λ1(ky) = ∆ejkyy1

(ky∆)2(1 + jky∆− ejky∆

)

ΛN (ky) = ∆ejkyyN

(ky∆)2(1− jky∆− e−jky∆

).

TEz Case

Piecewise linear expansion / Piecewise linear testing For piecewise constant func-

tions, we discretize the dielectric region into N=NxNy subsections of width ∆x = b/(Ny−1)

63

and ∆y = d/(Ny − 1), respectively, that approximate the original surface and approximate

J = xJx+ yJy by a linear combination of N piecewise linear basis functions with unknown

current coefficients Jxn , Jyn

Jx =N∑n=1

JxnΛn(x) (3.48a)

Jy =N∑n=1

JynΛn(y) (3.48b)

with Λn(x) and Λn(y) are given in (3.45) and (3.46), respectively. Substituting (3.48) into

(3.36) and testing with piecewise linear basis functions produces

N∑n=1

Jxn

jk0Z0

(1 +

1k20

∂2

∂x2

)

×xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′ +Λn(x)Λn(y)

jk0Y0 (εr(x, y) − 1)

+ Jyn

Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

= Eix(x, y)

(3.49a)

N∑n=1

Jxn

Z0

jk0

∂2

∂x∂y

xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

+ Jyn

jk0Z0

(1 +

1k20

∂2

∂y2

) xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

+Λn(x)Λn(y)

jk0Y0 (εr(x, y)− 1)

= Ei

y(x, y). (3.49b)

The impedance matrix elements shown in Appendix C can be obtained by evaluating the

coupled integrals in (3.49).

Piecewise constant expansion / Piecewise constant testing For piecewise constant

functions, we discretize the dielectric region into N=NxNy subsections of width ∆x = b/Nx

and ∆y = d/Ny, respectively, to approximate the original surface and represent J = xJx +

yJy by a linear combination of N piecewise constant basis functions with unknown current

64

coefficients Jxn , Jyn

Jz(x, y) =N∑n=1

JnΠn(x, y) =N∑n=1

JnΠn(x)Πn(y) (3.50)

with Πn(x) and Πn(y) are given in (3.38) and (3.39), respectively. Substituting (3.50) into

(3.36) and testing with piecewise linear basis functions produces

N∑n=1

Jxn

jk0Z0

(1 +

1k20

∂2

∂x2

)

×xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′ +Λn(x)Λn(y)

jk0Y0 (εr(x, y) − 1)

+ Jyn

Z0

jk0

∂2

∂x∂y

b2∫

− b2

d2∫

− d2

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

= Eix(x, y)

(3.51a)

N∑n=1

Jxn

Z0

jk0

∂2

∂x∂y

xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

+ Jyn

jk0Z0

(1 +

1k20

∂2

∂y2

) xn+∆x∫xn−∆x

yn+∆y∫yn−∆y

Λn(x′)Λn(y′)∑p,q

e−jkxp(x−x′) e−jkyq (y−y′)

k20 − k2

xp− k2

yq

dx′ dy′

+Λn(x)Λn(y)

jk0Y0 (εr(x, y)− 1)

= Ei

y(x, y). (3.51b)

The impedance matrix elements shown in Appendix C can be obtained by evaluating the

coupled integrals in (3.51).

3.3.3 Matrix Solution

In order to accurately determine the eigenvalues of the impedance matrices given in

Appendix C, a sufficient number of Floquet modes Np in the x direction and Nq in the y

direction and subsectional unknowns N=NxNy must be included. The minimization proce-

dure outlined in Chapter 2 is carried out for two-dimensional periodic media by fixing the

frequency of operation and one of the two propagation constants kx0 or ky0 and allowing the

optimization code to vary the other propagation constant until a minimum is found. If more

than one minimum (eigenvalue) is found, implying that multiple modes are propagating in

65

the structure, the corresponding eigenvectors are analyzed to determine which eigenvalue

corresponds to which propagating mode. A nontrivial solution for the fields requires the

matrix determinant to be zero, which results in a characteristic equation. The eigenvalues

(propagation constants) kx0 , ky0 are obtained from the roots of this equation. For a loss-

less structure, the propagation constant of a guided wave is a real number, however, in the

stopbands, the propagation constant is complex-valued.

In order to verify that the solution of the propagation constants computed using the

method of moments technique and PWE solution is correct, the band structure of a periodic

array of dielectric rods is calculated for a filling fraction b/a=d/c=0.3545 and relative di-

electric constant εr=8.9. Figure 3.9 shows the calculated band structure for both the TMz

and TEz cases. The moment method solution includes 225 subsectional bases (Nx=Ny=15)

and 961 Floquet modes (Np=Nq=31). The Fourier series solution uses 121 Floquet modes.

It is clear that the solutions obtained by the two methods are again in agreement with each

other and are also in agreement with [38, 129].

βx (βy=0) βy (βx=π/a) βx=βy

Normalized propagation constant ( βa)

0.00 3.14 6.28 9.42

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

TM (FS)TE (FS)TM (MoM)TE (MoM)

Figure 3.9: Band structure (TMz, TEz) of a periodic lattice of dielectric columns with fillingfraction b/a=d/c=0.3545, εr=8.9, Nx=Ny=15, and Np=Nq=31

Similar conclusions for two-dimensional dielectric structures as those expressed for one-

dimensional structures can be drawn about the number of unknowns needed to accurately

model the dielectric material and the number of requisite Floquet modes to include in the

double series to accurately determine the propagation constants. In Tables 2.1 – 2.4 of

66

Chapter 2, the accuracy of the propagation constant determined for both the TEx and

TMx cases increases both by increasing the number of Floquet mode contributions and

by modeling the equivalent current more carefully by increasing the number of unknowns.

A simple pulse basis expansion serves to solve the problem quickly and accurately. This

observation holds true as well for two-dimensional dielectric structures. As expected, the

accuracy of the propagation constants determined for both the TMz and TEz cases increases

with increasing Floquet contributions and with increasing subsectional bases. Additionally,

the advantage of incorporating higher order basis functions does not significantly increase

the accuracy and actually increases the cost and complication of numerical implementation.

Tables 2.1 – 2.4 in Chapter 2 clearly show that the accuracy of computing the propagation

constant is more sensitive to the number of subsectional bases used in the approximation

of the current than the number of Floquet modes used to compute the Floquet series. The

computational cost of computing each individual element in the impedance matrix in two

dimensions is proportional to NpNq. In two dimensions, the total cost of computing the full

impedance matrix is proportional to N2xN

2yNpNq. Thus, incorporating fewer basis functions

decreases the cost significantly.

As was mentioned previously in Chapter 2, the restriction that the IE/MoM solution

must compute the impedance matrix for each iteration of the optimizer requires significant

computational time. Consequently, for the band structure of an infinite lattice of dielectric

inserts in a background of differing dielectric constant, the solution obtained using the

Fourier series solution is the formulation of choice. The derivation and implementation of

the IE/MoM solution developed in Section 3.3 is extended to three-dimensional structures

in Chapter 4.

3.4 Effective Medium Theory (EMT)

To simplify the computation of the band structure for two-dimensionally periodic me-

dia, effective medium theory is applied to reduce the two-dimensional periodic structure

to a one-dimensional equivalent structure. This useful technique can be applied when the

period of the structure is much smaller than the wavelength. Using EMT, each periodic

row of the two-dimensional structure is replaced by a thin homogeneous layer of effective

permittivity [50] which is determined solely as a function of the geometrical and electrical

parameters of the lattice.

Consider the two-dimensional lattice with dielectric constant ε2 immersed in a back-

67

x

y

a

b

c d

ε eff

ε eff

ε eff

ε eff

ε2

ε1

Figure 3.10: Effective medium theory

ground with dielectric constant ε1 and the equivalent one-dimensional lattice with dielectric

constant εeff, illustrated in cross-section in Figure 3.10. Assume a plane wave is incident

along the x direction (Γ–X direction in the BZ). Effective medium theory allows for each

row of the square lattice to be replaced by an equivalent layer of permittivity εeff. For TMz

polarization, where the electric field parallel to z axis, Rytov [82] expands the effective

permittivity εeff in a power series of the period-to-wavelength ratio α=a/λ and is given by

εeff = ε0 +π2

3[f(1− f)(ε2 − ε1)]2 α2 +O(α4) (3.52)

where ε0 is the average relative permittivity ε0 = ε2f − ε1(1 − f), α = a/λ0, and f is the

filling fraction of the medium defined by the ratio of the width b of the effective homogeneous

layer to the lattice constant a. For TEz polarization, where the electric field parallel to y

axis, the effective permittivity is given by

εeff =1a0

+π2

3

[f(1− f)(ε2 − ε1)

ε2ε1

]2 ε0a3

0

α2 +O(α4) (3.53)

where a0 is the arithmetic average of the inverse relative permittivities a0 = f/ε2−(1−f)/ε1and f is defined above.

Two conclusions can be drawn about the results obtained using EMT. First, for large

period-to-wavelength ratios, the EMT approximation fails to accurately produce the band

structure for either the TMz or TEz case. This is expected since the physical boundary

conditions that exist in the real structure are not approximated well. Secondly, as the ratio

decreases, the validity of the EMT approximation increases. Even for a relatively large

period-to-wavelength ratio of 0.5, the results obtained using EMT are surprisingly good.

68

The band structure of the exact two-dimensional structure determined using the plane

wave expansion method and integral equation models outlined in Sections 3.2 and 3.3 and

the one-dimensional equivalent structure modeled using EMT is illustrated in Figure 3.11.

Thus, the lattice of Figure 3.10 with filling fraction b/a=d/c=0.3545 and dielectric constant

εr=8.9 is modeled by an array of dielectric slabs with the same filling fraction as the two-

dimensional array with effective permittivity εeff. For the TMz modes, the EMT results

compare favorably to [129] for normalized frequencies f0 < 1.0. The results obtained for

the TEz modes compares less favorably, particularly for higher frequencies.

If the filling fraction b/a=d/c is decreased to 0.2 and if all other geometrical parameters

remain the same, the ability of the EMT to effectively model the two-dimensional lattice as

an equivalent one-dimensional array can be seen in Figure 3.12 to improve greatly for the

TEz case. Although an improvement over the TMz case is not seen, the EMT effectively

models the first band gap as accurately as might be needed.

3.5 Conclusions

The solution for the propagation of electromagnetic energy through a two-dimensional

periodic dielectric structure has been determined using the plane wave expansion method

and an integral equation/method of moments solution. The plane wave expansion method

solves the two-dimensional (differential) wave equation by expanding the periodic functions

in a Fourier series. The integral equation solution is derived and implemented using polariza-

tion currents for a two-dimensionally periodic media. For infinite two-dimensional dielectric

structures, modeling the dielectric material with increasing number of unknowns increases

the accuracy of the solution as does increasing the number of Floquet modes. The disadvan-

tage of the IE/MoM solution is the significant increase in computational time required to

compute the eigenvalues (propagation constants) for the two-dimensional structures. The

advantage of the IE/MoM solution is its straightforward extension to three-dimensional

layered periodic structures with periodic material implants.

An effective medium theory (EMT) approximation for the two-dimensional lattice has

been shown to effectively model propagation in specific directions within the lattice struc-

ture. The use of EMT to effectively approximate propagation through a periodic structure

is significant since, for many of the applications used in microwave devices, the first (and

lowest) band structure is the band of interest. The approximation is particularly helpful

when one considers the complication of a two-dimensional formulation. Unfortunately, the

69

Normalized propagation constant (k a)

0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

0.75

1.00

TM modesTM EMT

(a)

Normalized propagation constant (k xa)

0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

0.75

1.00

TE modesTE EMT

(b)

Figure 3.11: Exact band structure and effective medium theory approximation for periodiclattice of dielectric columns with filling fraction b/a=d/c=0.3545, dielectricconstant εr=8.9, and period-to-wavelength ratio α = a/λ0=0.5 for (a) TMz

case and (b) TEz case

EMT is restricted to near normal incidence precluding its use for many applications of

interest.

70

Normalized propagation constant (k xa)

0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

0.75

1.00

TE modesTE EMT

(a)

Normalized propagation constant (k a)

0.00 1.57 3.14

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.00

0.25

0.50

0.75

1.00

TM modesTM EMT

(b)

Figure 3.12: Exact band structure and effective medium theory approximation for periodiclattice of dielectric columns with filling fraction b/a=d/c=0.2, dielectric con-stant εr=8.9, and period-to-wavelength ratio α = a/λ0=0.5 for (a) TMz caseand (b) TEz case

To validate the usefulness and realizability of microwave devices that incorporate sub-

strate materials with periodicities in two directions, the design, fabrication, and implemen-

71

tation of a two-dimensional periodic dielectric structure developed for use in parallel-plate

mode reduction in conductor-backed slots is presented extensively in Appendix D. Band

structures are designed to produce the requisite substrate properties using the PWE method

and IE/MoM solutions developed in this chapter.

72

CHAPTER 4

Scattering From an Inhomogeneous Doubly Periodic

Dielectric Layer Above a Layered Medium

4.1 Introduction

In this chapter, the solution found in [84] for the two-dimensional scattering from an

inhomogeneous periodic layer above a half-space layered medium is extended to general

three-dimensional scattering from an inhomogeneous doubly periodic layer above a half-

space layered medium. The solution of this problem is derived and implemented using a full-

wave integral equation/method of moments approach similar to the formulation for the one-

and two-dimensional periodic structures of Chapters 2 and 3. In this formulation, coupled

integral equations are derived that incorporate both the two-dimensional planar periodic

free-space Green’s function (PFSGF) and the inclusion of three-dimensional material blocks.

Equivalent polarization currents are used to model the material inclusions and the method

of moments is used to discretize the geometry and the integral equations. The resulting

matrix equation is numerically solved for various quantities of interest. First, the solution

of plane wave scattering from an inhomogeneous doubly periodic dielectric layer over a

layered medium is derived and implemented in Section 4.2. The reflection coefficient is

determined for arbitrary polarization (vertical, horizontal, or a combination thereof) and

arbitrary direction of incidence (φ0 ∈ [0, 2π], θ0 ∈ [0, π]) for a number of sample structures

and validated with results obtained from canonical problems and results published in the

open literature.

Of particular concern in the solution of the periodic structure is the convergence of

the resulting Floquet series. Although the contribution of the off-plane periodic elements

converges quickly, the convergence of the on-plane periodic elements is notoriously slow. In

73

order to compute the impedance matrix elements in a reasonable amount of time, various

series acceleration techniques and transformations have been suggested including the Pois-

son transformation [70], Kummer’s method [51], the Shanks’ transform [90], and the Ewald

transformation [30]. Summation acceleration techniques convert a slowly converging series

to a rapidly converging one by allowing the series to be transformed into a second series that

converges to the same limit but does so in a rapid fashion. The aforementioned transforms

have all been implemented in conjunction with the slowly-converging PFSGF and each

transform has its advantages and disadvantages when applied to the PFSGF in a particular

fashion. However, some of the transforms have only been implemented for limited cases

of the PFSGF. For the problems of interest in this chapter, a Poisson transformation and

a Shanks’ transformation are successfully implemented to improve both the speed and the

accuracy of the impedance matrix element computations. Additionally, Kummer’s method

is applied to specific series to compare the convergence rate of these different series accel-

eration techniques. A complete treatment of these and other series acceleration techniques

is found in Section 4.5.

4.2 Formulation of Integral Equations

In this section, volume integral equations (IE) are derived from Maxwell’s equations

and the boundary conditions that when solved yield equivalent currents from which total

scattered fields can be determined. The integral equations are discretized and cast into a

matrix equation form through the use of the method of moments (MoM) numerical solution

technique. The unknowns are obtained from the solution of the resulting linear system.

4.2.1 Derivation of Electric Field Integral Equation

The derivation of the electric field integral equation outlined in Chapter 2 is repeated

below for convenience. The total electric field is viewed as the sum of an incident field

Ei(r) due to radiation from a known source with the scatterer absent and a scattered field

E [J; r;V ] which is due to radiation by equivalent volume currents J which reside in a volume

V

E(r) = E [J; r;V ] + Ei(r), (4.1)

in which the operator E [J; r;V ] can be expressed in terms of a Hertz potential [108] as

E [J; r;V ] = k20 Π(r) +∇∇ ·Π(r) (4.2a)

74

where

Π(r) = − Z0

8π2k0

∫∫∫V

J(r′)∫∫∞

e−jkz|z−z′|

kze−jky(y−y′)e−jkx(x−x′) dkx dky dr′ (4.2b)

with

kz =

√k20 − k2

x − k2y k2

0 > k2x + k2

y,

−j√k2x + k2

y − k20 k2

0 < k2x + k2

y.(4.2c)

and Z0=1/Y0 is the intrinsic impedance of free-space. The electric field must satisfy

E [J; r;V ] + Ei(r) = E(r), r ∈ V. (4.3)

4.2.2 Integral Equations

Plane wave excitation

A representative grounded doubly periodic dielectric layer with period a in the x direc-

tion and period c in the y direction is illustrated in Figure 4.1. The perforated dielectric

a

cx

y

z

Figure 4.1: Doubly periodic dielectric layer with period a in the x direction and period cin the y direction over a layered medium

layer is excited by an incident time harmonic (ejωt) plane wave defined by

Ei = Pi e−jki·r =

(ehhi + evvi

)e−jki·r (4.4)

where Pi = ehhi+evvi denotes the polarization of the incident wave and ki is the direction

of propagation. The unit vectors hi, vi, and ki are defined in terms of the spherical angles

75

(φ0, θ0) as

ki = x cosφ0 sin θ0 + y sinφ0 sin θ0 − z cos θ0 (4.5a)

hi =ki × z

|ki × z| = x sinφ0 − y cosφ0 (4.5b)

vi = hi × ki = x cosφ0 cos θ0 + y sinφ0 cos θ0 + z sin θ0 (4.5c)

and are shown in Figure 4.2. The kDB coordinate system [45] consists of the wave vector

k and the plane containing the electric and magnetic flux density vectors D and B. In

z

y

x

φ

θ

hk

v

i

i

i

h s

vs ks

φ

θo

o

s

s

Figure 4.2: Coordinate system (kDB) for incident and scattered field

the absence of the dielectric layer (to be replaced by equivalent polarization currents), the

reflected field from a surface can be easily computed as

Er = Pre−jks·r =

(Rhehhs +Rvevvs

)e−jks·r (4.6)

where Rh and Rv are the Fresnel reflection coefficients for horizontal and vertical polarized

electric fields, respectively, and ks is the direction of propagation. The unit vectors hs, vs,

and ks are defined in terms of the spherical angles (φs, θs) as

ks = x cosφs sin θs + y sinφs sin θs + z cos θs (4.7a)

hs =ks × z

|ki × z| = x sinφs − y cosφs (4.7b)

vs = hs × ks = −x cosφs cos θs − y sinφs cos θs + z sin θs (4.7c)

76

Note that ks = ki − 2(z · ki)z.The total electric field everywhere is computed as the sum of the incident and reflected

electric fields in (4.4) and (4.6) and the scattered field Es(r) produced by the equivalent

induced electric current. Thus, (4.3) becomes upon substitution

J(r)jk0Y0 (εr(r)− 1)

= Ei(r) + Er(r) + Es(r)

=(ehhi + evvi

)e−jki·r +

(Rhehhs +Rvevvs

)e−jks·r + Es(x, y, z). (4.8)

The three coupled integral equations from (4.8) become

Jx(r)jk0Y0 (εr(r)− 1)

=(ehx · hi + evx · vi

)e−jki·r +

(Rhehx · hs +Rvevx · vs

)e−jks·r

+∫∫∫∞

Gxx(r; r′)Jx(r′) +Gxy(r; r′)Jy(r′) +Gxz(r; r′)Jz(r′)

dr′ (4.9)

Jy(r)jk0Y0 (εr(r)− 1)

=(ehy · hi + evy · vi

)e−jki·r +

(Rhehy · hs +Rvevy · vs

)e−jks·r

+∫∫∫∞

Gyx(r; r′)Jx(r′) +Gyy(r; r′)Jy(r′) +Gyz(r; r′)Jz(r′)

dr′ (4.10)

Jz(r)jk0Y0 (εr(r)− 1)

= evz · vie−jki·r +Rvevz · vse−jks·r

+∫∫∫∞

Gzx(r; r′)Jx(r′) +Gzy(r; r′)Jy(r′) +Gzz(r; r′)Jz(r′)

dr′ (4.11)

where Gxx—Gzz are the components of the dyadic free-space Green’s function.

Since the dielectric material is periodic in x with period a and prescribed phase shift

kx0a and periodic in y with period c and prescribed phase shift ky0c, the resulting equivalent

currents must satisfy

J(x+ pa, y + qc, z) = J(x, y, z)e−jkx0pae−jky0qc (4.12)

where

kx0 = k0 cosφ0 sin θ0

ky0 = k0 sinφ0 sin θ0

77

and p,q are the Floquet indices in the x - and y-directions, respectively. The x -component

of the scattered field integral, Ixx, in (4.9) is defined by

Ixx =∫∫∫∞

Gxx(r; r′)Jx(r′) dr′

=∑p,q

(p+ 12)a∫

(p− 12)a

(q+ 12)c∫

(q− 12)c

∞∫−∞

Gxx(r; r′)Jx(r′)e−jkx0pae−jky0qc dr′

=

a2∫

− a2

c2∫

− c2

∞∫−∞

Gpxx(r; r′)Jx(r′) dr′ (4.13)

where the periodic free-space Green’s function, denoted Gp(r, r′), is defined in terms of the

free-space Green’s function as

Gp(r; r′) =∑p,q

G(x, y, z;x′ + pa, y′ + qc, z′)e−jkx0pae−jky0qc. (4.14)

The significance of this procedure is that the solution of the periodic structure has now been

reduced to solving that of the single unit cell shown in Figure 4.3. Explicitly, Gpxx(r; r′) in

b

d

a

c

Figure 4.3: Doubly periodic dielectric layered unit cell

(4.13) can be written as

Gpxx(r; r′) = − Z0

8π2k0

∑p,q

(k20 +

∂2

∂x2

)∫∫∞

[e−jkz|z−z′|

kz+Rx

e−jkz(z+z′)

kz

]

× e−jky(y−y′−qc)e−jkx(x−x′−pa)e−jkx0pae−jky0qc dkx dky, (4.15)

78

where Rx = Rh. If the above is rewritten as

Gpxx(r; r′) = − Z0

8π2k0

∑p,q

(k20 +

∂2

∂x2

)∫∫∞

[e−jkz|z−z′|

kz+Rx

e−jkz(z+z′)

kz

]

× e−jkx(x−x′)e−jky(y−y′)e−j(kx0−kx)pae−j(ky0−ky)qc dkx dky, (4.16)

an important observation can be made which greatly simplifies the formulation. From the

theory of Fourier Series we can represent a periodic train of Dirac delta functions by an

infinite summation of complex exponentials,∑p

e−j(kx0−kx)pa =2πa

∑p

δ

[kx − kx0 −

2πpa

](4.17a)

∑q

e−j(ky0−ky)qc =2πc

∑q

δ

[ky − ky0 −

2πqc

]. (4.17b)

Substituting the Dirac delta functions representation in (4.17) and carrying out the inte-

grations in (4.16) yields

Gpxx(r; r′)

= − Z0

2k0ac

∑p,q

(k20 +

∂2

∂x2

)[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18a)

where

kxp = kx0 +2πpa

= k0 cosφ0 sin θ0 +2πpa

kyq = ky0 +2πqc

= k0 sinφ0 sin θ0 +2πqc

with

kzpq =

√k20 − k2

xp− k2

yq, k2

0 > k2xp

+ k2yq,

−j√k2xp

+ k2yq

− k20 , k2

0 < k2xp

+ k2yq.

The remaining periodic free-space Green’s function components Gpxy–Gpzz are defined by

Gpxy(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂x∂y

[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18b)

Gpxz(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂x∂z

[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18c)

79

Gpyx(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂y∂x

[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18d)

Gpyy(r; r′)

= − Z0

2k0ac

∑p,q

(k20 +

∂2

∂y2

)[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18e)

Gpyz(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂y∂z

[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18f)

Gpzx(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂z∂x

[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18g)

Gpzy(r; r′) = − Z0

2k0ac

∑p,q

∂2

∂z∂y

[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18h)

Gpzz(r; r′)

= − Z0

2k0ac

∑p,q

(k20 +

∂2

∂z2

)[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]e−jkxp(x−x′)e−jkyq (y−y′)

(4.18i)

where Ry = Rh and Rz = Rv. If the equivalent currents are assumed to radiate in the

presence of a ground plane, the Green’s functions listed in (4.18a)–(4.18i) can be easily

manipulated to agree with similar expressions in [133] by setting Rh=−1 and Rv=+1.

Reflection from a Dielectric Layer

If the ground plane backing the doubly periodic dielectric layer is replaced by a layered

medium, perhaps also grounded, the Fresnel reflection coefficients (Rh and Rv) are no

longer simply ±1. The geometry of the layered medium is shown in Figure 4.4 where the

80

region 1

region 2

region N

region 0

region (N+1)

θο

z=d

d

d

d

d

0

1

2

(N-1)

N

ε

ε

ε

ε

ε

0

1

2

N

(N+1)

E(H)

Figure 4.4: Plane wave reflection from layered medium

lth interface is the defined by the plane located at z=dl. Region 0 is the region above the

layered medium and region (N + 1) is the semi-infinite region below the N layers. The

solution for the total reflection coefficient of the medium can be obtained by applying the

appropriate boundary conditions to the tangential components of the electric and magnetic

field at each interface. Consequently, the (N + 1) interfaces yield (2N + 2) equations to

solve for the (2N + 2) unknown amplitude coefficients. Explicitly, the (2N + 2) unknowns

are the two unknowns in each of the N layers (one each for the waves traveling in the ±zdirections), one for the unknown transmission coefficient of the semi-infinite region, and one

for the reflection coefficient of interest.

Horizontal polarization Assume the electric field in the lth layer of a medium is polar-

ized in the horizontal direction. From (4.4) and following [84], the electric field in the lth

layer is written in the form

Ehl=[cile

jkzlz + crl e

−jkzlz]e−jkx0xe−jky0y (4.19)

where

kzl= k0

√εl − sin2 θ0 (4.20)

and cil and crl are the amplitudes of the −z and +z traveling waves in the lth layer, respec-

tively, and εl is the permittivity of the lth layer. The magnetic field in the lth layer can

be found by applying (2.1b) to the electric field above. Carrying out the required vector

operations, one finds that the tangential components of the vertically polarized magnetic

81

field in the lth layer have the form

Hvl=

kzl

k0Z0

[cile

jkzlz − crl e−jkzl

z]e−jkx0xe−jky0y. (4.21)

The following recursive relationship that relates the amplitudes of the lth layer to those of

the (l + 1)th layer can be derived by requiring the tangential electric and magnetic field to

be continuous across each of the dielectric interfaces

crlcil

=

(crl+1/c

il+1

)+ Γhl e

−j2kz(l+1)dl(

crl+1/cil+1

)Γhl + e

−j2kz(l+1)dle−j2kzl

dl (4.22)

where

Γhl =kzl

− kz(l+1)

kzl+ kz(l+1)

. (4.23)

The solution for the total horizontal reflection coefficient Rh is initiated by assuming

ci0=1 in region 0 and by noting that crN+1=0 in the semi-infinite region. Starting from

cr(N+1)/ci(N+1)=0, equations (4.22) and (4.23) are used repeatedly to solve for the amplitudes

of the waves in each layer beginning with the interface located at z = dN and continuing to

the interface between the zeroth and first layer. The horizontal reflection coefficient is then

determined as Rh=cr0/ci0.

Vertical polarization If the electric field is vertically polarized, the same recursive re-

lation in (4.22) is found but with Γhl replaced by

Γvl =ε(l+1)kzl

− εlkz(l+1)

ε(l+1)kzl+ εlkz(l+1)

. (4.24)

The solution for the vertical reflection coefficient is found from Rv=cr0/ci0.

Grounded medium If the N th layer in the medium is grounded, ΓhN=−1 and ΓvN=+1,

and the process is executed as before.

4.3 Numerical Implementation

The first step in the method of moments (MoM) numerical method [34] is to discretize

the geometry and approximate the unknown electric current in the dielectric region with

either subsectional or entire domain basis functions. The equivalent polarization currents

are expanded in a linear combination of these basis functions, and the equations are tested

82

in order to obtain an adequate number of equations to solve for the unknown coefficients of

the basis functions.

The dielectric region is discretized into N=NxNyNz piecewise constant subvolumes of

width ∆x = b/Nx, length ∆y = d/Ny, and depth ∆z = t/Nz. The current J = xJx +

yJy + zJz is expanded in a linear combination of N piecewise constant basis functions

with unknown current coefficients Jxn , Jyn , Jzn centered at xn = − b2 + ∆x(n − 1

2), yn =

−d2 +∆y(n− 1

2), and zn = ∆z(n− 12)

Jx =N∑n=1

JxnΠn(x, y, z) =N∑n=1

JxnΠn(x)Πn(y)Πn(z) (4.25a)

Jy =N∑n=1

JynΠn(x, y, z) =N∑n=1

JynΠn(x)Πn(y)Πn(z) (4.25b)

Jz =N∑n=1

JznΠn(x, y, z) =N∑n=1

JznΠn(x)Πn(y)Πn(z) (4.25c)

where

Πn(x) =

1, xn − ∆x

2 < x < xn + ∆x2 ;

0, otherwise;(4.26)

and

Πn(y) =

1, yn − ∆y

2 < y < yn +∆y

2 ;

0, otherwise;(4.27)

and

Πn(z) =

1, zn − ∆z

2 < z < zn + ∆z2 ;

0, otherwise.(4.28)

4.3.1 Impedance Matrices

The scattered field is determined by integrating the equivalent currents over the periodic

Green’s function. By discretizing the resulting coupled equations, they can be cast into the

following matrix form Zxxmn Zxymn Zxzmn

Zyxmn Zyymn Zyzmn

Zzxmn Zzymn Zzzmn

Jxn

Jyn

Jzn

=

V xm

V ym

V zm

. (4.29)

83

Specifically, for the Zxx submatrix, the current representation is inserted into (4.9) and

the integrations are carried out. Testing the resulting equation using point collocation, we

obtain for all off-plane, i.e. zm = zn, elements

Zxx =

a2∫

− a2

c2∫

− c2

t∫0

Gpxx(r; r′) Jx(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

(k20 +

∂2

∂x2

)[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jx(x′, y′, z′) dx′ dy′ dz′ (4.30)

Zxxmn = −Z0∆x∆y∆z

2k0ac

∑p,q

(k20 − k2

xp

kzpq

)[e−jkzpq |zm−zn| +Rxe−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.31)

For zm = zn, the integration over z must be computed carefully. The resulting on-plane

impedance matrix elements for Zxx are

Zxxmn = −Z0∆x∆y∆z

2k0ac

∑p,q

(k20 − k2

xp

kzpq

)

×[(2− 2e−jkzpq∆z/2

)kzpq∆z

+Rx sinc(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn). (4.32)

The remaining impedance matrix elements Zxymn–Zzzmn are explicitly below given for on-plane

and off-plane elements.

Submatrix Zxy

Zxy =

a2∫

− a2

c2∫

− c2

t∫0

Gpxy(r; r′) Jx(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂x∂y

[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jx(x′, y′, z′) dx′ dy′ dz′ (4.33)

84

For zm = zn,

Zxymn =Z0∆x∆y∆z

2k0ac

∑p,q

kyqkxp

kzpq

[e−jkzpq |zm−zn| +Rye−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.34)

For zm = zn,

Zxymn =Z0∆x∆y∆z

2k0ac

∑p,q

kyqkxp

kzpq

[(2− 2e−jkzpq∆z/2)

kzpq∆z+Ry sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.35)

Submatrix Zyx

Zyx =

a2∫

− a2

c2∫

− c2

t∫0

Gpyx(r; r′) Jy(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂y∂x

[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jy(x′, y′, z′) dx′ dy′ dz′ (4.36)

For zm = zn,

Zyxmn =Z0∆x∆y∆z

2k0ac

∑p,q

kyqkxp

kzpq

[e−jkzpq |zm−zn| +Rxe−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.37)

For zm = zn,

Zyxmn =Z0∆x∆y∆z

2k0ac

∑p,q

kyqkxp

kzpq

[(2− 2e−jkzpq∆z/2)

kzpq∆z+Rx sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.38)

85

Submatrix Zyy

Zyy =

a2∫

− a2

c2∫

− c2

t∫0

Gpyy(r; r′) Jy(x′, y′, z′) dx′ dy′ dz′

= −k0Z0

2ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

(1 +

1k20

∂2

∂y2

)[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jy(x′, y′, z′) dx′ dy′ dz′ (4.39)

For zm = zn,

Zyymn = −Z0∆x∆y∆z

2k0ac

∑p,q

(k20 − k2

yq

kzpq

)[e−jkzpq |zm−zn| +Rye−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.40)

For zm = zn,

Zyymn = −Z0∆x∆y∆z

2k0ac

∑p,q

(k20 − k2

yq

kzpq

)

×[(2− 2e−jkzpq∆z/2

)kzpq∆z

+Ry sinc(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.41)

Submatrix Zxz

Zxz =

a2∫

− a2

c2∫

− c2

t∫0

Gpxz(r; r′) Jx(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂x∂z

[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jx(x′, y′, z′) dx′ dy′ dz′ (4.42)

86

For zm = zn,

Zxzmn =Z0∆x∆y∆z

2k0ac

∑p,q

kxp

[sgn (zn − zm)e−jkzpq |zm−zn| +Rze−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.43)

For zm = zn,

Zxzmn =Z0∆x∆y∆z

2k0ac

∑p,q

kxp

[Rz sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.44)

Submatrix Zzx

Zzx =

a2∫

− a2

c2∫

− c2

t∫0

Gpzx(r; r′) Jz(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂z∂x

[e−jkzpq |z−z′|

kzpq

+Rxe−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jz(x′, y′, z′) dx′ dy′ dz′ (4.45)

For zm = zn,

Zzxmn =Z0∆x∆y∆z

2k0ac

∑p,q

kxp

[sgn (zn − zm)e−jkzpq |zm−zn| +Rxe−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.46)

For zm = zn,

Zzxmn =Z0∆x∆y∆z

2k0ac

∑p,q

kxp

[Rx sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.47)

87

Submatrix Zyz

Zyz =

a2∫

− a2

c2∫

− c2

t∫0

Gpyz(r; r′) Jy(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂y∂z

[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jx(x′, y′, z′) dx′ dy′ dz′ (4.48)

For zm = zn,

Zyzmn =Z0∆x∆y∆z

2k0ac

∑p,q

kyq

[sgn (zn − zm)e−jkzpq |zm−zn| +Rze−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.49)

For zm = zn,

Zyzmn =Z0∆x∆y∆z

2k0ac

∑p,q

kyq

[Rz sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.50)

Submatrix Zzy

Zzy =

a2∫

− a2

c2∫

− c2

t∫0

Gpzy(r; r′) Jz(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

∂2

∂z∂y

[e−jkzpq |z−z′|

kzpq

+Rye−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jz(x′, y′, z′) dx′ dy′ dz′ (4.51)

For zm = zn,

Zzymn =Z0∆x∆y∆z

2k0ac

∑p,q

kyq

[sgn (zn − zm)e−jkzpq |zm−zn| +Rye−jkzpq (zm+zn)

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.52)

88

For zm = zn,

Zzymn =Z0∆x∆y∆z

2k0ac

∑p,q

kyq

[Ry sinc

(kzpq∆z

2

)e−jkzpq2zm

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.53)

Submatrix Zzz

Zzz =

a2∫

− a2

c2∫

− c2

t∫0

Gpzz(r; r′) Jz(x′, y′, z′) dx′ dy′ dz′

= − Z0

2k0ac

a2∫

− a2

c2∫

− c2

t∫0

∑p,q

(k20 +

∂2

∂z2

)[e−jkzpq |z−z′|

kzpq

+Rze−jkzpq (z+z′)

kzpq

]

× e−jkxp(x−x′)e−jkyq (y−y′) Jz(x′, y′, z′) dx′ dy′ dz′ (4.54)

For zm = zn,

Zzzmn = −Z0∆x∆y

2k0ac

∑p,q

sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

×[ zn+∆z

2∫zn−∆z

2

(k20 − ∂2

∂z∂z′

)e−jkzpq |zm−z′|

kzpq

dz′ +

zn+∆z2∫

zn−∆z2

(k20 +

∂2

∂z2

)Rze−jkzpq (zm+z′)

kzpq

dz′]

(4.55)

Carrying out the integrations in (4.55) leads to the following off-plane elements of the Zzzmnsubmatrix

Zzzmn = −Z0∆x∆y∆z

2k0ac

∑p,q

(k20 − k2

zpq

)[e−jkzpq |zm−zn|

kzpq

+Rze−jkzpq (zm+zn)

kzpq

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)sinc

(kzpq∆z

2

)e−jkxp(xm−xn)e−jkyq (ym−yn). (4.56)

For zm = zn, particular care must be taken to evaluate (4.54). If in the computation the on-

plane matrix elements for the Zzz submatrix the second derivative in (4.54) is not evaluated

carefully, erroneous results will be produced [76]. From distribution theory, carrying out

the inner derivative of the first integral in (4.55) yields

∂

∂z′e−jkzpq |z−z′| = jkzpq sgn(z − z′)e−jkzpq |z−z′|. (4.57)

89

where

sgn(z − z′) =

+1, z > z′;

−1, z < z′.(4.58)

Applying the outer derivative on the result obtained in (4.57) gives

∂

∂z

sgn(z − z′)e−jkzpq |z−z′|

=[−jkzpq + 2δ(z − z′)] e−jkzpq |z−z′|. (4.59)

Consequently,

∂2

∂z∂z′e−jkzpq |z−z′| = jkzpq

[−jkzpq

+ 2δ(z − z′)]e−jkzpq |z−z′|

=[k2zpq

+ 2jkzpqδ(z − z′)]e−jkzpq |z−z′|. (4.60)

Inserting the above into (4.54) and evaluating the first integral produces

Zzzmn = −Z0∆x∆y

2k0ac

∑p,q

sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

×[Rz

(k20 − k2

zpq

) e−j2kzpqzm

kzpq

∆z sinc(kzpq∆z

2

)+ k2

0

zn+∆z2∫

zn−∆z2

e−jkzpq |z−z′|

kzpq

dz′

−zn+∆z

2∫zn−∆z

2

[k2zpq

+ 2jkzpqδ(z − z′)] e−jkzpq |z−z′|

kzpq

dz′]. (4.61)

Rewriting the above as

Zzzmn = −Z0∆x∆y

2k0ac

∑p,q

sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

×[Rz

(k20 − k2

zpq

) e−j2kzpqzm

kzpq

∆z sinc(kzpq∆z

2

)+(k20 − k2

zpq

) (2− 2e−jkzpq∆z/2)

k2zpq

−zn+∆z

2∫zn−∆z

2

2jkzpqδ(z − z′)e−jkzpq |z−z′|

kzpq

dz′]

(4.62)

and carrying out the remaining integration yields the on-plane matrix elements for the Zzz

submatrix

Zzzmn = −Z0∆x∆y

2k0ac

∑p,q

sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

×[Rz

(k20 − k2

zpq

) e−j2kzpqzm

kzpq

∆z sinc(kzpq∆z

2

)+(k20 − k2

zpq

) (2− 2e−jkzpq∆z/2)

k2zpq

−2j]

(4.63)

90

A similar expression to (4.63) can be found by using dyadic analysis [45] and performing

some algebraic manipulation.

4.3.2 Excitation Matrices

The excitation matrix can be found by testing the incidence excitation using point

collocation. For plane wave excitation,

V xm = −

(ehx · hi + evx · vi

)e−jki·rm −

(Rhehx · hs +Rvevx · vs

)e−jks·rm (4.64a)

V ym = −

(ehy · hi + evy · vi

)e−jki·rm −

(Rhehy · hs +Rvevy · vs

)e−jks·rm (4.64b)

V zm = −evz · vie−jki·rm −Rvevz · vse−jks·rm (4.64c)

4.4 Results

A number of simple checks can be performed that illustrate the validity of the formula-

tion and the correctness of the implementation. A grounded dielectric layer can be modeled

using the method of moments solution by setting the filling fractions b/a and d/c to one

(1) and setting the horizontal and vertical reflection coefficients for the ground plane to −1

and +1, respectively, independent of the incidence angle. The moment method solution for

the plane wave scattering from the grounded periodic dielectric structure yields identical

results with the analytical solution obtained for the plane wave scattering of a grounded

layer. The exact horizontal and vertical reflection coefficients at the surface of the grounded

dielectric layer of thickness t and relative permittivity εr are

Rh =kz0 sin kz1t+ jkz1 cos kz1tkz0 sin kz1t− jkz1 cos kz1t

ej2kz0t (4.65a)

Rv =εrkz0 cos kz1t− jkz1 sin kz1tεrkz0 cos kz1t+ jkz1 sin kz1t

ej2kz0 t (4.65b)

where kz0=k0 cos θ0 and kz1=k0√εr − sin2 θ0. Figure 4.5 shows the calculated phase angle

of the reflection coefficient as a function of incidence angle θ0 using the method of moments

solution for 81 subsectional unknowns (Nx=Ny=Nz=3 for each of the three polarizations),

arbitrary plane of incidence φ0=45, unit cell size a=0.1λ0, relative permittivity εr=2.56,

and dielectric thickness t=0.15λd where λd=λ0/√εr. The solutions were determined using

at most 3721 Floquet modes (Np=Nq=61). Also included in the figure is the exact analytical

solution for the two polarizations. The reflection coefficients are evaluated at a height

z0=20t above the ground plane and theoretically should have a magnitude of one (1) for

91

θο (deg)

0.00 30.00 60.00 90.00

Phas

e (d

eg)

-180

-90

0

90

180

RhRvRh (MoM)Rv (MoM)

Figure 4.5: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of θ0 forNx=Ny=Nz=3, Np=Nq=61, and a=0.1λ0

the grounded case. For the data shown in Figure 4.5 where the dielectric is lossless, the

magnitudes of Rh and Rv are 1.00±0.08. The accuracy of the phase values obtained for

Rv degrades slightly at grazing angles near θ0=90. As the angle of incidence increases,

the z -variation in the dielectric cannot be modeled accurately using only 3 unknowns. If

the z -variation is modeled using five (5) or seven (7) unknowns, the error in the magnitude

decreases to less than 2% for all angles of incidence.

Additionally, for the grounded dielectric layer, the value of the reflection coefficients

should be independent of both the plane of incidence and the unit cell size. Identical phases

for the reflection coefficients are calculated for a number of different values of φ0. However,

when the unit cell size is increased, additional unknowns in x - and y-directions must be

used to model the material blocks in order to obtain comparable accuracies to cases where

the unit cell size is smaller. It is interesting to note that for the grounded dielectric layer,

the scattered field produced by the equivalent currents for the two polarizations is exactly

180 out-of-phase. Although the magnitudes and phases of the currents induced by the two

polarizations are quite different, the scattered electric field produced by them is shifted in

phase by π (Rh=−Rv for the ground plane).

As the filling fraction varies from one (1) to zero (0) and unit cell spacings less than a

wavelength, the phase angle of the horizontal and vertical reflection coefficients determined

92

using the method of moments solution should vary between the exact value obtained from

a grounded dielectric layer and the exact value obtained from a ground plane alone. This

is shown in particular in Table 4.1 for a grounded dielectric layer of relative permittivity

εr=2.56, dielectric thickness t=0.15λd, φ0=45, θ0=45, and a=0.25λ0 at z0=20t using 192

subsectional unknowns (Nx=Ny=Nz=4 for each of the three polarizations) and Np=Nq=61.

The exact values for b/a=1 and b/a=0 shown in bold in the table are calculated using the

analytical solution of (4.65).

Table 4.1: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for Nx=Ny=Nz=4, Np=Nq=61, φ0=45, θ0=45, and a=0.25λ0

b/a Rh Rv

1.00 -64.99 89.52

0.90 -63.42 95.15

0.75 -59.90 103.77

0.50 -56.56 115.48

0.25 -55.01 123.33

0.00 -54.59 125.41

The grounded dielectric slab can also be modeled as a grounded layered medium as

shown in Figure 4.4. This is accomplished by replacing the homogeneous layer of thick-

ness t and relative dielectric constant εr with two layers each of thickness t/2. Region 2

is filled with a homogeneous layer of permittivity εr, region 1 is replaced with equivalent

polarization currents as before, and the ground is located at z=d2. The significant differ-

ence in computation for the layered medium case is the calculation of Rh and Rv. Unlike

the previous example, where the dielectric is replaced by equivalent polarization currents

and the scattering occurs from the ground plane, the reflection coefficients for equivalent

currents radiating over a layered dielectric are, in general, functions of the complex an-

gle, γpq=arctan(kρ/kzpq) where kρ is the phase constant in the transverse plane equal to√k2xp

+ k2yq. Consequently, Rh and Rv must be calculated for every combination of Floquet

indices p, q.The phase angle of the reflection coefficient for the grounded two-layer medium is de-

termined in Figure 4.6 for the same parameters as Figure 4.5. The noticeable increase in

the accuracy of the phase obtained in this solution is attributable to the increased accuracy

93

θο (deg)

0.00 30.00 60.00 90.00

Phas

e (d

eg)

-180

-90

0

90

180

RhRvRh (MoM)Rv (MoM)

Figure 4.6: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of θ0 forNx=Ny=Nz=3, Np=Nq=61, and a=0.1λ0 calculated using the multilayeredsolution

in modeling the dielectric region. The same number of unknowns are used to model the

dielectric layer of thickness t/2 as were used to model the dielectric layer of thickness t in

Figure 4.5. For all angles of incidence, the magnitudes of Rh and Rv are 1.00±0.02. If

the polarization currents in the previous example are arbitrarily set to zero (J=0) or if

the relative permittivity is allowed to approach one (εr → 1) and the solution for the total

scattered field is recomputed, the reflection coefficients should be equivalent to those of a

grounded layer of thickness t/2. This is found to be true in particular for θ0=45 where the

phase of Rh and Rv for both the exact and calculated solution are −55.72 and +110.01,

respectively.

4.5 Series Acceleration Techniques for the Periodic, Free-

Space Green’s Function

Unfortunately, the series listed throughout Section 4.3 have serious but well-documented

convergence issues when z is very close or equal to z′ that must be addressed before an accu-

rate and rapid solution can be found. Summation acceleration techniques convert a slowly

converging series to a rapidly converging one by allowing the series to be transformed into

94

a second series that converges to the same limit but does so in a rapid fashion. A number

of series acceleration techniques have been applied to the periodic, free-space Green’s func-

tion (PFSGF) to accelerate the notorious on-plane convergence problem [42]. Some of the

more useful ones are listed in the Table 4.2. The transformations listed in the first part of

the table are ones implemented in this work; those listed in the second part of the table

are ones that are implemented elsewhere. Each acceleration technique has advantages and

disadvantages and must be applied carefully to the series of interest in order to obtain the

desired results of improved accuracy and increased speed.

Table 4.2: Summary of series transformations

Transformations

Poisson transformation

Shanks’ transformation

Wynn’s ε-algorithm

Kummer transformation

Ewald transformation

ρ-algorithm

θ-algorithm

Chebyshev-Toeplitz (CT) algorithm

Levin’s t-transform

4.5.1 Acceleration Techniques Applied in This Work

Poisson Transformation [70, 51, 99]

The Poisson summation converts a slowly converging series to a rapidly converging one

by allowing the series to be summed in the Fourier transform domain. The reciprocal

spreading property of the Fourier transform says that narrower support in one domain

would necessarily require wider support in the other domain. The transformation can be

written as∞∑

n=−∞f(t+ nT ) =

1T

∞∑n=−∞

e−jnω0tF (nω0) ω0 =2πT. (4.66)

The Poisson transformation transforms the PFSGF into its spectral form which makes it

amenable for use in spectral domain analysis (SDA) of layered media or periodically loaded

95

dielectrics.

To obtain a summation for a two-dimensional array of point current sources, the Poisson

summation formula is first applied to the y coordinate of the three-dimensional Green’s

function

∑p,q

f(p, q) =∑p,q

[(x− pa)2 + (y − qc)2 + z2]1/2 e−jk[(x−pa)2+(y−qc)2+z2]1/2

=∑p,q

12πc

K0

[(2πq

c

)2

− k20

]1/2 [(x− pa)2 + z2]1/2

e−j 2πq

cy. (4.67)

An expression equivalent to a two-dimensional Green’s function can be recovered by ma-

nipulation of the above expression giving

∑p,q

f(p, q) =∑p,q

14cjH

(2)0

([k2 −

(2πqc

)]1/2 [(x− pa)2 + z2]1/2

)e−j

2πqcy. (4.68)

Applying the Poisson summation formula again, but this time to the x coordinate gives the

following Poisson summation formula for the Green’s function f(p, q):

∑p,q

f(p, q) =12ac

∑p,q

e−|z|

[( 2πp

a )2+( 2πqc )2−k2

]1/2

√(2πpa

)+(

2πqc

)− k2

e−j2πp

axe−j

2πqcy. (4.69)

Equation (4.70) becomes upon substitution of a phase shift k = kx0 x+ ky0 y

∑p,q

f(p, q) =1

2jac

∑p,q

e−jkzpq |z|√k2 −

(2πpa

)2+(

2πqc

)2e−jkxpxe−jkyqy (4.70)

Shanks’ Transform [90, 99, 101, 43]

For alternating series, the partial sums of a sequence can be treated as a mathematical

transient of the sum S of the series. Assume the nth partial sum is

Sn = S +n∑

k=0

αkqnk

for |qk| < 1. Hence, Sn → S as n→ ∞. The sum S can be found by expressing Sn in terms

of S, α, and q for indices (n− 1), n, and (n + 1),

Sn = S + αqn

Sn−1 = S + αq(n−1)

Sn+1 = S + αq(n+1)

96

and solving for the sum S. The sum is found to be equal to

S =Sn+1Sn−1 − S2

n

Sn+1 + Sn−1 − 2Sn. (4.71)

Two simple examples will illustrate that the sum S of simple alternating series can be

determined without summing an inordinate number of terms.

Example 1. Geometric series The geometric series

∞∑k=0

qk = 1 + q + q2 + · · · (4.72)

converges to sum 1/(q − 1) if |q| < 1 and αk = 1k. Setting S0 = 1, S1 = 1 + q, and

S2 = 1 + q + q2, and solving for S using (4.71) yields

=(1 + q + q2)− (1 + q)2

1 + q + q2 + 1− 2(1 + q)=

−q−q + q2

=1

q − 1.

Example 2. Alternating geometric series

∞∑k=0

(−1)kqk = 1− q + q2 − · · · (4.73)

converges to sum 1/(q + 1) if |q| < 1 and αk = (−1)k. Setting S0 = 1, S1 = 1 − q, andS2 = 1− q + q2, and solving for S using (4.71) yields

S =(1− q + q2)− (1− q)2

1− q + q2 + 1− 2(1− q) =q

q2 + q

=1

q + 1.

Another effective transform for alternating series is the Shanks’ transform. The algo-

rithm for the Shanks’ transform of a sequence of partial sums is

S = e1(Sn+1) +1

e1(Sn+1)− e1(Sn) , (4.74)

where

e0(Sn) = Sn, e1(Sn) =1

e0(Sn+1)− e0(Sn) .

97

Higher order Shanks transforms can be carried out using Wynn’s ε-algorithm [124, 15, 101]:

es+1(Sn) = es−1(Sn+1) +1

es(Sn+1)− es(Sn) , s = 1, 2, · · · (4.75)

where

e0(Sn) = Sn, e1(Sn) =1

e0(Sn+1)− e0(Sn)Only the even order terms e2r(Sn) are Shanks’ transforms of order r approximating S. The

process is applied continually until a desired criterion is reached. For the series used in this

work, the following convergence criterion was used. The convergence factor εc is defined by∣∣∣∣ek(Sn−k)− ek−2(Sn−k+2)ek(Sn−k)

∣∣∣∣ ≤ εc. (4.76)

In order to assure that the summation has adequately converged, the algorithm is contin-

ued until three successive values of ek+2(Sn−k−2), ek+4(Sn−k−4), and ek+6(Sn−k−6) satisfy

(4.76).

Table 4.3: Convergence of higher order Shanks’ transforms of Leibnitz series as a functionof order and number, π=3.14159265358979 . . .

n e0 = Sn e2 e4 e6 e8 e10

0 4.0000000 3.1666667 3.1423423 3.1416149 3.1415933 3.1415926(7)

1 2.6666667 3.1333333 3.1413919 3.1415873 3.1415925

2 3.4666667 3.1452381 3.1416667 3.1415943 3.1415927

3 2.8952381 3.1396825 3.1415634 3.1415921

4 3.3396825 3.1427128 3.1416065 3.1415967

5 2.9760462 3.1408813 3.1415854

6 3.2837385 3.1420718 3.1415929

7 3.0170718 3.1412548

8 3.2523659 3.1415927

9 3.0418396

10 3.2323158

Example. Leibnitz series

π = 4∞∑n=0

(−1)n

2n+ 1= 3.14159265358979 . . . (4.77)

98

Table 4.3 shows the results of applying various order Shanks’ transforms to Leibnitz series.

One notes that using a fourth-order Shanks’ transform is accurate to six significant digits

but only requires the computation of 10 terms in the series and 40 floating point operations

(FLOPs). A direct summation of the series would require calculating and summing 25×106

terms to achieve a similar accuracy of five significant digits.

The Shanks’ transform is simple and efficient – the first N terms of the series are the

only terms computed. The higher order transforms are applied to the N terms using the

simple algorithm listed above. The total number of floating point operations required to

compute a (N, k) order Shanks’ transform can be calculated to be

FLOPs =k∑

n=0

(N − n) = N(k + 1)−k∑

n=1

n

=(N − k

2

)(k + 1). (4.78)

In Table 4.4, the error and relative computation time for various combination order Shanks’

transformations on (4.77) are shown. The baseline computation time is determined by

Table 4.4: Error for various (N, k) Shanks’ transforms for Leibnitz series

N k ek(SN ) Error FLOPs

10 0 3.23231580 9.07 × 10−2

50 0 3.16119861 1.96 × 10−2

100 0 3.15149340 9.90 × 10−3

10 2 3.14173610 1.43 × 10−4 27

10 4 3.14159331 6.58 × 10−7 40

10 6 3.14159266 5.11 × 10−9 49

10 8 3.14159265 5.43× 10−11 54

50 2 3.14159443 1.78 × 10−6 147

50 4 3.14159265 5.97× 10−10 240

100 2 3.14159443 2.36 × 10−7 297

assuming that 100 terms of the sum are computed. The relative computational time for

the various order transforms are computed by applying the transforms to different num-

bers of terms in the series. Because the determination of the higher order transforms are

computationally inexpensive, it is efficient to set k=N -2. When k=N -2, the error can be

minimized without increasing the computational cost significantly. The total number of

99

FLOPs required for a (N ,N -2) order Shanks’ transform is

FLOPs =N−2∑n=0

(N − n) =(N

2+ 1)(N − 1) N2

2(4.79)

For a double-sided series, the number of FLOPs required is simply twice the number for a

single-sided series and can be approximated byN2. For example, a (10,4) Shanks’ transform

of (4.77) results in an error of 6.58×10−7 and requires 80 FLOPs. However, simply carrying

out two more transforms (requiring 28 more FLOPs) yields an error of only 5.43 × 10−11.

Also, note that a (10,8) transform yields a significantly smaller error than a (100,2) trans-

form and does so at a significantly reduced computational cost. For more realistic series

where calculating the individual terms is expensive, the determination of the minimum num-

ber of terms N and the subsequent order k=N -2 of the transform to accurately compute

the sum is imperative.

(30,28) Shanks’ transform of inner sum of Zyymm impedance matrix element

An example of the significant increase in accuracy and decrease in computational cost that

implementing a (N ,N -2) Shanks’ transform produces is easily seen by computing the inner

sum for the (m,n) element of the Zyy submatrix in (4.41) and implementing a (30,28)

Shanks’ transform. In a double summation, such as those found in this work, the Shanks’

transform is applied to the sequence of partial sums over the inner indices for a specific

outer index. The transform is then applied to the outer sequence. For convenience, the Zyy

submatrix is repeated below

Zyymn = −k0Z0∆x∆y

2ac

∑p

∑q

(1− k2

yq

k20

)

×[2j − 2je−jkzpq ∆z/2

k2zpq

+Ry(γpq)∆z sinc(kzpq∆z

2

)e−j2kzpqzm

kzpq

]

× sinc(kxp∆x

2

)sinc

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn) (4.80)

The slowly converging part of the sum in (4.80) is the first expression. Simplifying (4.80)

by assuming that Ry = 0, xm = xn, ym = yn, and zm = zn yields an equation whose

convergence rate is relatively unchanged by the simplification. The resulting equation is

Zyy = −2k0Z0

ac

∑p

sin(kxp∆x/2

)kxp

∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyq

[2j − 2je−jkzpq∆z/2

k2zpq

].

(4.81)

100

FLOPs

102 103 104 105 106 107

Rel

ativ

e er

ror

10-8

10-7

10-6

10-5

10-4

10-3

10-2Direct sumShanks (12,10)Shanks (16,14)Shanks (20,18)Shanks (24,22)Shanks (28,26)

Figure 4.7: Relative error as a function of floating point operations for computing variousorder Shanks’ transforms and the direct sum of the inner sum of (4.80)

The inner sum is evaluated assuming the index of the outer sum p = 0. It is clear that

implementing a (N ,N -2) Shanks’ transform to sum the series minimizes the relative error for

a given computational cost. The relative error as a function of computational cost (FLOPs)

of implementing various order Shanks’ transforms is shown in Figure 4.7. The number of

floating point operations for the direct sum of the series is also included for comparison.

Notice the dramatic decrease in the relative error of the sum for a given number of FLOPS

required to sum the inner sum of (4.80) to a given accuracy by implementing different

(N ,N -2) Shanks’ transforms. To achieve a relative error of 10−4 for the inner sum, only

103 FLOPS are required by the Shanks’ transform, whereas 104 FLOPS are needed by the

direct sum. The disparity in computational cost between computing the direct sum and

implementing a Shanks’ transform is even more significant for a relative error of less than

10−6.

If (4.80) is used without simplification, the series that must be summed is

Zyy = −2k0Z0

ac

∑p

sin(kxp∆x/2

)kxp

e−jkxp(xm−xn)

×∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyq

e−jkyq (ym−yn)

[2j − 2je−jkzpq ∆z/2

k2zpq

](4.82)

101

However, the same Shanks’ transform is used without modification to sum this element.

This is significant in light of the fact that all of the sums can be computed using the same

general subroutine. This is not true for other summation acceleration techniques.

Kummer’s Method [51, 99]

The statement that the rate of convergence of a series is determined by the asymptotic

form of the series is the basis for Kummer’s method. Suppose a series f(n) has an asymptotic

form f∞(n). An equivalent expression to f(n) can be found if the asymptotic series is

subtracted from and added back to the original series as seen below∞∑

n=−∞f(n) =

∞∑n=−∞

[f(n)− f∞(n)] +∞∑

n=−∞f∞(n). (4.83)

Usually, f∞(n) is chosen in such a way that the last series has a known closed-form expres-

sion. For complicated series, however, obtaining a closed-form expression for the asymptotic

series can be a tedious task.

The double summation that is found in this work is written symbolically as

S =∞∑m=0

∞∑n=0

Smn.

The asymptotic series Amn is subtracted from and added to the original series Smn

S =∞∑m=0

∞∑n=0

(Smn −Amn) +∞∑n=0

Amn

.

If the asymptotic series Amn has a closed-form solution in terms of the inner index n equal

to Am, then the above equation can be written as

S =∞∑m=0

∞∑n=0

(Smn −Amn) +Am.

The sum Smn −Amn converges much more rapidly than the direct sum of Smn.

Additional series acceleration can be achieved for some series by incorporating a similar

technique on the tail contribution of the series. Substituting Zmn = Smn−Amn and writing

the series Zmn as the sum of (N + 1) terms and the remaining tail contribution yields

S =∞∑m=0

N−1∑n=0

Zmn +∞∑

n=N

Zmn +Am

.

The asymptotic tail series Tmn of the tail contribution is subtracted from and added to the

original series Zmn much like before

S =∞∑m=0

N−1∑n=0

Zmn +∞∑

n=N

(Zmn − Tmn) +∞∑

n=N

Tmn +Am

.

102

The tail contribution is now written as∑∞

n=N Tmn =∑∞

n=0 Tmn−∑N−1

n=0 Tmn and the above

equation can be written as

S =∞∑m=0

N−1∑n=0

Zmn +∞∑

n=N

(Zmn − Tmn) +∞∑n=0

Tmn −N−1∑n=0

Tmn +Am

.

If Tm is the closed-form solution for the inner sum of Tmn, then

S =∞∑m=0

N−1∑n=0

Zmn +∞∑

n=N

(Zmn − Tmn) + Tm −N−1∑n=0

Tmn +Am

.

The sum Zmn − Tmn also converges much more rapidly than the direct sum of Zmn.

Kummer’s method applied to inner sum of Zyymm

An example of the usefulness of Kummer’s method is seen in computing the same inner sum

for the (m,n) element of the Zyy submatrix of (4.41) as was determined using a Shanks’

transform. Beginning with (4.80), the inner sum is computed using Kummer’s method.

Only one expression of the inner series is not exponentially convergent:

S′p = 2j

∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyqk

2zpq

(4.84)

where S′p is the sum over all q excluding the following q = 0 term

Sp0 =

(1− k2

y0

k20

)sin (ky0∆y/2)

ky0

[2j − 2je−jkzp0∆z/2

k2zp0

].

For large q, the sum in (4.84) can be approximated by

S′p = −2j

k20

′∑q

2πqc

1

k20 − k2

xp−(

2πqc

)2 sin(kyq∆y/2

)

=2jk20

′∑q

2πqc

( c2π

)2 1

q2 −[(

ck02π

)2 −(ckxp

2π

)2] sin (kyq∆y/2

)(4.85)

with

kyq 2πqc, k2

zpq k2

0 − k2xp

−(2πqc

)2

.

If we represent the denominator as q2 − α2p, where α2

p = (ck0/2π)2 − (ckxp/2π

)2, thenS′p =

cj

πk20

′∑q

q

q2 − α2p

sin(kyq∆y/2

). (4.86)

103

Noting that kyq = ky0 +2πqc and using a simple trigonometric identity, (4.86) can be written

as

S′p =

cj

πk20

′∑q

q

q2 − α2p

[sin (ky0∆y/2) cos (qπ∆y/c) + cos (ky0∆y/2) sin (qπ∆y/c)

]. (4.87)

The sums in (4.87) have asymptotic forms that can be evaluated analytically. If we rewrite∑′q as

∑−1q=−∞+

∑∞q=1 and observe that

q

q2 − α2p

cos qζ = − q

q2 − α2p

cos(−qζ)q

q2 − α2p

sin qζ =q

q2 − α2p

sin(−qζ)

then (4.87) can be rewritten as

S′p =

2cjπk2

0

cos (ky0∆y/2)∞∑q=1

q

q2 − α2p

sin qζ (4.89)

where ζ = π∆y/c. The sum in (4.89) has an exact solution given by

∞∑q=1

q sin qζq2 + α2

p

=π

2sinhαp(π − ζ)

sinhαpπ, α2

p > 0, 0 < ζ < 2π (4.90a)

∞∑q=1

q sin qζq2 − α2

p

=π

2sinαp(π − ζ)

sinαpπ, α2

p > 0, 0 < ζ < 2π. (4.90b)

The asymptotic value S∞p of (4.86) is

S∞p =

cj

k20

cos (ky0∆y/2)sinαp(π − ζ)

sinαpπ.

Since Sp = Sp0 + S′p + S∞

p ,

Sp =∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyq

[2j − 2je−jkzpq ∆z/2

k2zpq

]+cj

k20

cos (ky0∆y/2)sinαp(π − ζ)

sinαpπ

− 2cjπk2

0

cos (ky0∆y/2)∞∑q=1

q

q2 − α2p

sin qζ (4.91)

As can be seen in Figure 4.8, this method also significantly reduces the relative error for a

given computational cost. Included for reference is the Shanks’ transform and direct sum

data of Figure 4.7. A similar convergence rate as the Shanks’ transform is found using

Kummer’s method. However, even this simple series requires concentrated analytical effort.

104

FLOPs

102 103 104 105 106 107

Rel

ativ

e er

ror

10-8

10-7

10-6

10-5

10-4

10-3

10-2Direct sumShanks (12,10)Shanks (16,14)Shanks (20,18)Shanks (24,22)Shanks (28,26)Kummer

Figure 4.8: Relative error as a function of floating point operations for computing variousorder Shanks’ transforms, Kummer’s transforms, and the direct sum of the innersum of (4.80)

If (4.41) is used without simplification, i.e. Ry = 0, xm = xn, ym = yn, but zm = zn,

the series that must be summed is

Zyy = −2k0Z0

ac

∑p

sin(kxp∆x/2

)kxp

e−jkxp(xm−xn)

×∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyq

e−jkyq (ym−yn)

[2j − 2je−jkzpq ∆z/2

k2zpq

]. (4.92)

Implementing the above sum using a Shanks’ transform is easily done by applying the

general subroutine to the first (N +1) terms of the series. However, a new analytic solution

must be determined when implementing Kummer’s method. The term of the inner series

that is not exponentially convergent is

S′p =

2jk20

∑q

(k20 − k2

yq

)kyqk

2zpq

sin(kyq∆y/2

)e−jkyq (ym−yn) (4.93)

where S′p is the sum over all q excluding the following q = 0 term

Sp0 =

(1− k2

y0

k20

)sin (ky0∆y/2)

ky0e−jky0(ym−yn)

[2j − 2je−jkzp0∆z/2

k2zp0

].

105

For large q, the sum in (4.93) can be approximated by

S′p = −2j

k20

′∑q

2πqc

1

k20 − k2

xp−(

2πqc

)2 sin(kyq∆y/2

)e−jkyq (ym−yn)

=2jk20

′∑q

2πqc

( c2π

)2 1

q2 −[(

ckxp

2π

)2 −(ck02π

)2] sin (kyq∆y/2

)e−jkyq (ym−yn) (4.94)

with

kyq 2πqc, k2

zpq k2

0 − k2xp

−(2πqc

)2

.

If we represent the denominator as q2 − α2p, where α2

p = (ck0/2π)2 − (ckxp/2π

)2, thenS′p =

cj

πk20

′∑q

q

q2 − α2p

sin(kyq∆y/2

)e−jkyq (ym−yn) (4.95)

Similar to before, (4.95) can be written as

S′p =

cj

πk20

e−jky0(ym−yn) sin (ky0∆y/2)′∑q

q

q2 − α2p

cos (qπ∆y/c) e−j2πq

c(ym−yn)

+cj

πk20

e−jky0(ym−yn) cos (ky0∆y/2)′∑q

q

q2 − α2p

sin (qπ∆y/c) e−j2πq

c(ym−yn). (4.96)

The sums in (4.96) have asymptotic forms that can be evaluated analytically with a little

analytical effort. Equation (4.96) can be rewritten as

S′p =

cj

πk20

e−jky0(ym−yn) sin (ky0∆y/2)∞∑q=1

q

q2 − α2p

[−j sin q(ζ+) + j sin q(ζ−)

]

+cj

πk20

e−jky0(ym−yn) cos (ky0∆y/2)∞∑q=1

q

q2 − α2p

[sin q(ζ+) + sin q(ζ−)

](4.97)

where ζ± = π∆y [1± 2(m− n)] /2. Using (4.90), the asymptotic value of (4.97) can be

written as

S∞p =

c

2k20

e−jky0(ym−yn) sin (ky0∆y/2)[sinαp(π − ζ+)

sinαpπ− sinαp(π − ζ−)

sinαpπ

]

+cj

2k20

e−jky0(ym−yn) cos (ky0∆y/2)[sinαp(π − ζ+)

sinαpπ+

sinαp(π − ζ−)sinαpπ

]. (4.98)

106

Again, Sp = Sp0 + S∞p + S′

p,

Sp =∑q

(1− k2

yq

k20

)sin(kyq∆y/2

)kyq

e−jkyq (ym−yn)

[2j − 2je−jkzpq ∆z/2

k2zpq

]

− cj

πk20

e−jky0(ym−yn) sin (ky0∆y/2)∞∑q=1

q

q2 − α2p

[−j sin q(ζ+) + j sin q(ζ−)

]

− cj

πk20

e−jky0(ym−yn) cos (ky0∆y/2)∞∑q=1

q

q2 − α2p

[sin q(ζ+) + sin q(ζ−)

]

+c

2k20

e−jky0(ym−yn) sin (ky0∆y/2)[sinαp(π − ζ+)

sinαpπ− sinαp(π − ζ−)

sinαpπ

]

+cj

2k20

e−jky0(ym−yn) cos (ky0∆y/2)[sinαp(π − ζ+)

sinαpπ+

sinαp(π − ζ−)sinαpπ

](4.99)

As can be clearly seen, the analytical effort to determine the asymptotic form of the elements

in the impedance matrix is not trivial; the requisite computer coding to implement this

solution lends itself to difficult debugging. Additionally, applying Kummer’s method to the

inner sum does not produce an analytical solution for the outer sum. Thus, one must either

compute an addition analytical solution for the outer sum or find another mechanism to

compute the sum. In light of the difficulties found in computing realizable double sums,

implementing a general Shanks’ transform may be more profitable.

Another application of Kummer’s method in the acceleration of the PFSGF series is

to simply sum the series incorporating a “smoothing” or “acceleration” parameter in the

propagation constant. The asymptotic form of the new “modified” Green’s function is then

subtracted from the original series and the transformed series is added back [40].

4.5.2 Acceleration Techniques Used Elsewhere

Ewald Transformation [30, 39, 65, 27]

The Ewald transformation expresses the periodic free-space Green’s function as the sum

of two doubly infinite series, namely one series summed in the spectral domain and one series

summed in the spatial domain. Similarly to Kummer’s method, the Ewald transformation

casts the PFSGF into a hybrid form more amenable to acceleration. The Ewald transfor-

mation results in series that utilizes the complementary error function yielding two more

rapidly converging series.

107

ρ-algorithm [124, 102]

For monotonic series, the ρ-algorithm is a very rapidly converging series acceleration

technique. However, it does not fare as well with alternating series. The simple ρ-algorithm

can be computed as follows

ρ(n)k = ρ(n+1)

k−2 +k

ρ(n+1)k−1 − ρ(n)

k−1

, k = 1, 2, · · · (4.100)

where

ρ(n)−1 = 0, ρ

(n)0 = Sn (4.101)

and where k is the order of the algorithm. The even order terms, ρ(n)2k , give the estimate of

the sum.

θ-algorithm [15, 104]

Another rapidly accelerating series technique for alternating series is the θ-algorithm.

The θ-algorithm can be computed as follows

θ(n)2k+2 = θ(n+1)

2k +

[θ(n+2)2k − θ(n+1)

2k

] [θ(n+2)2k+1 − θ(n+1)

2k+1

][θ(n+2)2k+1 − 2θ(n+1)

2k+1 + θ(n)2k+1

] , k = 0, 1, 2, · · · (4.102)

and the odd order terms by

θ(n)2k+1 = θ(n+1)

2k−1 +1[

θ(n+1)2k − θ(n)

2k

] , k = 1, 2, · · · (4.103)

where

θ(n)−1 = 0, θ

(n)0 = Sn (4.104)

The even order terms, θ(n)2k+2, give the estimate of S. Like the ρ-algorithm, the θ-algorithm

is easy to implement.

108

Chebyshev-Toeplitz algorithm [121, 105]

The Chebyshev-Toeplitz (CT) algorithm can be computed as follows

t(n)−1 = 0, t

(n)0 = Sn, σ0 = 1, (4.105)

t(n)1 = t(n)

0 + 2t(n+1)0 , σ1 = 3, (4.106)

t(n)k+1 = 2t(n)

k + 4t(n+1)k − t(n)

k−1, k = 1, 2, · · · (4.107)

σk+1 = 6σk − σk−1, k = 1, 2, · · · (4.108)

T(n)k =

t(n)k

σk, k = 0, 1, 2, · · · (4.109)

The nth iterate of the CT algorithm is given by T (n)k which gives as estimate of the sum of

the series.

Levin t-transform [54, 106]

The t-transform algorithm can be computed as follows

t(n)k =

∑ki=0(−1)i

(ki

)(n+in+k

)(k−1)(

Sn+i

Sn+i−1−Sn+i

)∑k

i=0(−1)i(ki

)(n+in+k

)(k−1)(

1Sn+i−1−Sn+i

) , k = 1, 2, · · · (4.110)

The nth iterate of the Levin t-transform is given by T (n)k which gives an estimate of sum of

the series.

4.6 Conclusions

A full-wave IE/MoM solution that determines the general three-dimensional scattering

from an inhomogeneous doubly periodic layer above a half-space medium has been for-

mulated, implemented, and validated. Equivalent volume polarization currents are used

to replace the dielectric material and the combination of a Poisson transformation and a

Shanks’ transformation is used to improve the speed and accuracy of the impedance matrix

element computations. The accuracy of the solution is primarily dependent on three fac-

tors: (i.) representing the dielectric material with an appropriate number of subsectional

unknowns, (ii.) including enough terms in the resulting Floquet summations, and (iii.)

the choice of an appropriate series acceleration convergence technique. A general rule-of-

thumb in MoM formulations is to use 5–10 subsectional unknowns per half wavelength in

the medium. This is the minimum number that should be used when modeling the z varia-

tion of the material. Fewer unknowns per wavelength are required to model the transverse

109

variation to obtain a given accuracy because the fields are more smoothly varying in the

transverse direction.

The solution procedure is valid for reasonable combinations of unit cell spacing, material

shape, dielectric constant, and number of layers. This solution procedure is subsequently

used in Chapter 5 to validate the modeling of an equivalent uniaxial layer that replaces the

doubly periodic layer for use in rectangular patch antenna applications.

110

CHAPTER 5

Radiation Properties of a Rectangular Microstrip Patch on a

Uniaxial Substrate

5.1 Introduction

Of interest to the applied microwave community is the implementation of a microstrip

patch antenna on a doubly periodic dielectric layered medium. Preliminary investigations

of simple Hertzian dipoles radiating over this structure have been addressed by Yang [133]

using an analytic array scanning technique [67]. However, the solution in [133] is compu-

tationally intensive and has not been extended for arbitrary polarization. A simpler (and

perhaps more insightful) solution where the doubly periodic dielectric layer is emulated by

an equivalent anisotropic (uniaxial) material is developed in the following sections. The

uniaxial model for the doubly periodic dielectric layer is validated using plane wave reflec-

tion coefficients for a variety of filling fractions, angles, and permittivities. Although the

solution of a similar approximation has been developed for two-dimensional scattering from

periodicities in one-dimension [85, 87], this is the first known treatment of such a solution

for full three-dimensional scattering from dielectric periodicities in two directions.

Traditionally, microstrip patch antennas have been integrated on relatively low permit-

tivity substrates in order to improve antenna performance. Since microstrip patches are

effectively a radiating resonator, the large Q produced by confining the stored energy in a

thin region under the patch necessarily narrows the bandwidth significantly. Typical band-

widths for traditional patch designs are on the order of 2–4%. New designs for microstrip

patch feeds have increased bandwidths but at the cost of increased complexity. Integrating

the antenna on higher permittivity substrates is preferred to minimize circuit size and spu-

rious radiation [75, 89] but at the cost of confining the potential radiating energy even more

111

tightly. This trade-off between good antenna performance and good circuit performance is

a key design feature found in many microstrip antenna designs. The compromise can be

achieved primarily through the proper design of any or all of the three main components in a

microstrip structure: radiating element, feed mechanism, and substrate choice. (Extensive

bibliographies can be found in [75, 89].) The focus of this chapter is the choice and design of

an effective anisotropic substrate for use in microstrip antenna applications. Additionally,

the solution for a rectangular patch element radiating over the equivalent uniaxial substrate

is carried out accurately and efficiently using Ansoft’s High Frequency Structure Simulator

that incorporates anisotropic substrates.

In order to reduce the often unwanted formation of surface waves which can lead to pat-

tern degradation and low efficiencies, relatively thin homogeneous substrates (thicknesses

less than 0.05λ0) are used. However, as noted before, this limitation has severe conse-

quences. Although a good deal of attention has focused on integrating microstrip patches

on homogeneous substrates, many of the practical substrates in use today such as sapphire,

Epsilam-10, and boron nitride, have a significant amount of (uniaxial) anisotropy. The

uniaxial materials have two effective homogeneous permittivities: one permittivity aligned

parallel to the optical axis of the material and one permittivity aligned perpendicular to the

optical axis [47, 85, 87]. Rigorous full-wave solutions, albeit computationally intensive ones,

have been developed to characterize the effect of anisotropy on various patch antenna param-

eters such as resonant length and efficiency [74, 68, 131]. Other investigations [19, 14, 123]

have focused on supplementing the conclusions drawn in earlier investigations, namely, that

the primary effect of anisotropy on rectangular patch antennas is the change in its resonant

length. This is significant because of the narrow bandwidth of the patch itself. The rel-

atively large shift in resonant frequency produced in many of the modern substrates may

actually force a rectangular patch designed to operate at a specific frequency to radiate out-

side of the antenna bandwidth [74]. Additionally, anisotropic effects are found that shape

the radiation pattern of the patch and thus in an array configuration, the coupling to other

elements.

Because uniaxial substrates are often expensive to manufacture and have limited flexibil-

ity for design, uniaxial substrates can be emulated (and easily fabricated) by incorporating

periodic inclusions in an otherwise homogeneous substrate. In particular, if the wavelength

is sufficiently large compared with the period of the array, the doubly periodic dielectric layer

(in the transverse direction) can emulate a uniaxial substrate with an optical axis aligned

in a direction normal to the layer [85, 87]. The periodic structure shown in Figure 5.1(a)

112

can be easily constructed using simple milling or etching techniques from simple inexpen-

ε

ε

ε0

2

1

(a)

ε

ε

ε0

2

t , εz

(b)

Figure 5.1: (a) Periodic dielectric layer over a layered medium and (b) equivalent uniaxialmedium with transverse permittivity εt and axial permittivity εz over a layeredmedium

sive, homogeneous substrates. This is significant because some common uniaxial materials

such as sapphire that are expensive to grow [6] can be “artificially” replicated easily and

inexpensively. Additionally, the artificial nature of the periodic uniaxial substrate per-

mits the creative design of new substrates with the expanded freedom of anisotropic ratio,

background permittivity, and/or fabrication technique.

5.2 Equivalent Uniaxial Modeling

The equivalent uniaxial material of Figure 5.1(b) with the optical axis aligned with the

z axis has an assumed permittivity tensor of

¯ε = ε0

εx 0 0

0 εy 0

0 0 εz

(5.1)

where εx=εy=εt is the transverse component and εz is the component along the direction

of the optical axis.

Two of the well-known dielectric mixing models1 for two-phase media (i.e., ε is piecewise

constant and takes on only two distinct values each) are the Hashin-Shtrikman bounds [35,

47] and the Licktenecker bounds [25]. As with many of the early approximations, these

solutions are determined using variational methods, and thus, the solutions take on the

form of bounds for the equivalent two-phase materials (permittivities, conductivities, or1These models belong to a more general class of dielectric mixing formulas that include the Maxwell-

Garnet mixing formula (accurate for relatively low volume fractions) and the Polder van Santen mixingformula (accurate for relatively low permittivity changes).

113

permeabilities). For the Hashin-Shtrikman and the Licktenecker models outlined below,

both the unit cell and the material inclusions are assumed to have square2 cross-sections.

5.2.1 Hashin-Shtrikman Model

A good approximation (assuming the inclusions are small with respect to wavelength) for

the longitudinal component of the permittivity tensor, εz, is simply to take the volumetric

averages of the constituent materials and is given by

εz = (1− f2)εa + f2εb (5.2)

where f is the filling fraction, εa is the background permittivity, and εb is the permittivity of

the material blocks or inclusions. The generalized Hashin-Shtrikman formulas provide upper

and lower bounds for the transverse component of the permittivity tensor, εt [48, 47]. While

no known closed form expression exists for the transverse permittivity, evidence suggests

that the lower and upper Hashin-Shtrikman bounds provide good accuracy to approximate

the value over a wide range of parameters for many practical substrates. The lower and

upper bounds, denoted εLHS and εUHS , respectively, are defined as

εLHS = εa2εb + f2 (εa − εb)2εa + f2 (εb − εa) (5.3a)

εUHS = εb(εa + εb) + f2 (εa − εb)(εa + εb) + f2 (εb − εa) . (5.3b)

5.2.2 Licktenecker Model

Another set of bounds that serves to approximate the effective transverse permittivity

can be derived using the Licktenecker bounds [25]. The Licktenecker bounds can be written

as ∫ b

0

dy∫ a0

dxε(x,y)

≤ εx ≤∫ a

0

dx∫ b0

dyε(x,y)

(5.4a)

∫ a

0

dx∫ b0

dyε(x,y)

≤ εy ≤∫ b

0

dy∫ a0

dxε(x,y)

. (5.4b)

However, if the inclusion is symmetric about the line x=y, the bounds take on simpler

forms since εx=εy=εt. Carrying out the integrals in (5.4), the lower and upper Licktenecker2Similar results can be derived for circular inclusions in a square unit cell but are not presented. For

electrically small cells, the particular geometric shape is insignificant compared to the volumetric averagesof the constituent phases.

114

bounds, denoted εLL and εUL , respectively, are determined to be

εLL =fεb + (1− f)εa

fεa + f(1− f)εb + (1− f)2εa (5.5a)

εUL =fεb + f(1− f)εa + (1− f)2εb

fεa + (1− f)εb (5.5b)

where f is the filling fraction b/a, εa is the background permittivity, and εb is the permittiv-

ity of the material blocks or inclusions. Note that if the phasing of the material is reversed,

1/εLL = εUL .

The effective transverse permittivity of the two-dimensional periodic media determined

by the Hashin-Shtrikman and Licktenecker bounds is graphed as a function of filling fraction

b/a for εb = 10εa in Figure 5.2. Also included in the figure is the solution obtained from

f

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ε t

1

2

3

4

5

6

7

8

9

10εz

εHSL

εHSU

εLL

εLU

Figure 5.2: Effective transverse permittivity as a function of filling fraction for εb=10εa

simply weighting the constitutive phases by their respective volume fractions. Note that

this is nothing more than the effective permittivity in the axial direction determined from

(5.2). Consequently, the assumption that the permittivity of the two dimensional periodic

substrate (even for electrically small cells) has an effective homogeneous permittivity leads

to erroneous results. Additionally, one observes that the upper Hashin-Shtrikman bound

is grossly in error. However, this error has been reported previously in the literature [47].

The other three bounds are reasonably precise and have been reported to have small errors

of less than 4%. The solutions obtained for the Licktenecker bounds are precise, and since

they are derived using specific information about the geometry of the inclusions, one can

115

conclude these bounds are more accurate. Overall, the precision of the three remaining

bounds is found to increase with increasing filling fraction.

f

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ε t

1

2

3

4

5

6

7

8

9

10εz

εHSL

εHSU

εLL

εLU

Figure 5.3: Effective transverse permittivity as a function of filling fraction for εa=10εb

If the phases of the two materials are reversed as shown in Figure 5.3, a noticeable

change in precision can be seen. The upper Hashin-Shtrikman bound is again seen to have

significant error. The remaining three bounds are reasonably close for large filling fractions,

but have some noticeable differences at lower filling fractions. Again, the incorporation

of geometrical features necessarily makes the Licktenecker bounds a more accurate solu-

tion. Consequently, a good approximation for the effective transverse permittivity can be

determined by taking the geometric mean of εLL and εUL

εt =√εLLε

UL . (5.6)

Not only are the values obtained using the geometric mean of εLL and εUL remarkably close

to the values yielded by the lower Hashin-Shtrikman bound for the phase set in Figure 5.2,

but also for the second case shown in Figure 5.3 when the phases are reversed.

An interesting feature of the emulated anisotropic material is the ability to achieve a

desired anisotropic ratio (AR) by incorporating either a periodic dielectric region in an air

background or by including periodic air inclusions in a dielectric background depending on

the application and ease of fabrication. The AR is a function of the dielectric contrast

between the material blocks or inclusions and the background medium. If the effective

116

transverse permittivity εt is greater than the effective axial permittivity εz (εt ≥ εz), the

material is said to have negative uniaxial anisotropy (AR > 1). Conversely, if the effec-

tive axial permittivity εz is greater than the effective transverse permittivity εt (εt ≤ εz),

the material is said to have positive uniaxial anisotropy (AR < 1). For example, in the

microwave region, sapphire (a common substrate material used in microwave integrated

circuits) has an AR of 0.81 (εt=9.4, εz=11.6) [6]. Thus, sapphire has positive uniaxial

anisotropy. Boron nitride has an AR of 1.5 (εt=5.12, εz=3.4) and Epsilam-10 has an AR

of 1.26 (εt=13.0, εz=10.3) [6]. Hence, boron nitride and Epsilam-10 have negative uniaxial

anisotropy. Most of the soft composite ceramic-impregnated teflon-type substrates, such

as Rogers’ RT/Duroid 6000 series, have anisotropic ratios greater than one. The physical

significance of the AR will be addressed in the next section.

For the substrates with periodicities in the plane of the substrate, the emulated material

is necessarily a positive uniaxial material. Since εt and εz are known quantities, specifying

one of the three unknowns, namely εa, εb or f , one can solve for the remaining two unknowns

of interest from (5.2) and (5.5). For example, sapphire can be emulated by a medium

consisting of periodic dielectric rods with a dielectric constant of 12.1 and filling fraction

equal to 0.95, or by a medium consisting of periodic air inclusions in a dielectric background

with dielectric constant of 16.0 and filling fraction equal to 0.3 as seen in Figure 5.4. The

second medium can be viewed more intuitively as doubly periodic set of dielectric veins

a

b

ε2ε1

b = 0.95 aε = 12.1 ε2 1

(a)

a

b

ε2

ε1

b = 0.30 aε = 16.0 ε2 1

(b)

Figure 5.4: Uniaxial nature of sapphire modeled by (a) periodic dielectric rods withε2=12.1ε1 and b/a=0.95 and (b) periodic dielectric veins with ε2=16.0ε1 andb/a=0.30

117

in an air background. This observation is consistent with the two-dimensional periodic

structures developed in Chapter 3 and detailed in Section 3.2.2.

As was introduced earlier, the most significant application for the emulated uniaxial

materials is the variety of artificial substrate designs available to the RF designer. Naturally

occurring uniaxial materials have fixed anisotropic ratios and permittivity tensors and limit

important parameter choices. However, depending on the application and the materials

available for use, a particular AR can be be emulated. This flexibility allows the RF designer

not only the choose of a particular AR, but also the choice of the dielectric constant of the

background medium and/or inclusions.

Using the Hashin-Shtrikman or Licktenecker bounds, one can compute the minimum

achievable AR of the equivalent positive uniaxial medium by differentiating the ratio εt/εz

with respect to the filling fraction f where εt is found using (5.3b) or (5.5) and εz is found

from (5.2). For electrically small cells, the minimum AR that can be achieved for a given

dielectric constant ratio, either for εb/εa > 1 or εb/εa < 1, is shown in Table 5.1. Note

Table 5.1: Minimum achievable anisotropic ratio (AR)

(εb/εa) Min AR εt εz b/a

1.00 1.00 1.00 1.00 1.00

2.32 0.90 1.45 1.61 0.46

4.00 0.74 1.74 2.35 0.45

6.15 0.59 1.94 3.28 0.44

10.20 0.43 2.13 5.00 0.44

(εa/εb) Min AR εt εz b/a

1.00 1.00 1.00 1.00 1.00

2.32 0.93 1.48 1.58 0.56

4.00 0.85 1.87 2.18 0.61

6.15 0.79 2.23 2.81 0.65

10.20 0.73 2.76 3.78 0.70

that the minimum achievable AR is strongly dependent on the permittivity of the dielectric

contrast between the phases. Thus, in order to emulate a given permittivity set (εt,εz),

the ratio of the relative dielectric constant of the two phases must be large enough to both

produce the minimum achievable AR listed in Table 5.1 and to also achieve the value of

118

the desired axial component of the permittivity tensor. As will be seen in the next section,

the AR is a parameter that has a significant effect on the resonant length of a radiating

structure located on or near such a material.

In order to verify that using the previous models are valid approximations for the uni-

axial layer, the reflection coefficient for the equivalent uniaxial medium is compared to the

IE/MoM numerical solution of the reflection coefficient obtained from the doubly periodic

grounded dielectric layer modeled using polarization currents. Thus, the exact horizontal

and vertical reflection coefficients at the surface of the grounded uniaxial dielectric layer of

thickness t with dielectric permittivity tensor given in (5.1) must be found.

5.3 Plane Wave Reflection Coefficients for a Grounded Uni-

axial Layer

Assuming the equivalent uniaxial layer has the permittivity tensor ¯ε given in (5.1) and

and starting from Maxwell’s equations for a uniaxial medium

∇× E = −jωµH (5.7)

∇× H = jω ¯ε ·E, (5.8)

one can solve for the wave equations that must be satisfied in the medium. Following [74]

the wave equation for the z -component of the electric field can be determined to be

∂2

∂x2Ez +

∂2

∂y2Ez +

εzεt

∂2

∂z2Ez + εzk2

0Ez = 0 (5.9)

where k0=ω√µ0ε0=2π/λ0. The wave equation for the z -component of the magnetic field

can also be determined and is found to be

∂2

∂x2Hz +

∂2

∂y2Hz +

∂2

∂z2Hz + εtk2

0Ez = 0. (5.10)

The dispersion relations for (5.9) and (5.10) can be found by assuming waves of the form

e±jkxxe±jkyye±jkzz. Thus, for TEz (transverse electric to z ) or horizontal polarized waves,

k2z = εtk2

0 − β2 = k2a (5.11)

where β2 = k2x + k2

y and ka is the z -component of the propagation vector. For TMz (trans-

verse magnetic to z ) or vertical polarized waves,

k2z = εtk2

0 − εtεzβ2 = k2

b (5.12)

119

where kb is the z -component of the propagation vector. Note that the uniaxial material

produces two different propagation vectors in the medium for the two polarizations.

We now define the Fourier transform pair

E(x, y, z) =∫∫∞

E(kx, ky, z)ejkzzejkyy dkx dky (5.13a)

E(kx, ky, z) =1

4π2

∫∫∞

E(x, y, z)e−jkzze−jkyy dx dy. (5.13b)

In the transform domain, the transverse fields can be written in terms of Ez and Hz as(εtk

20 +

∂2

∂z2

)Ex = jkx

∂

∂zEz + ωµ0kyHz (5.14a)(

εtk20 +

∂2

∂z2

)Ey = jky

∂

∂zEz − ωµ0kxHz (5.14b)(

εtk20 +

∂2

∂z2

)Hx = jkx

∂

∂zHz − ωε0εtkyEz (5.14c)(

εtk20 +

∂2

∂z2

)Hy = jky

∂

∂zHz + ωε0εtkxEz (5.14d)

where ∂2

∂z2= −k2

z and kz is either ka for TEz or kb for TMz waves. Using the dispersion

relations in (5.11) and (5.12), the above equations can be simplified to

Ex =jkxεzεtβ2

∂

∂zEz +

ωµ0kyβ2

Hz (5.15a)

Ey =jkyεzεtβ2

∂

∂zEz − ωµ0kx

β2Hz (5.15b)

Hx =jkxβ2

∂

∂zHz − ωε0εzky

β2Ez (5.15c)

Hy =jkyβ2

∂

∂zHz +

ωε0εzkxβ2

Ez. (5.15d)

Extending [74], if we assume solutions for Ez and Hz in region 1 have the form

Ez = ejkz0z +Ae−jkz0z (5.16a)

Hz = ejkz0z +Be−jkz0z (5.16b)

and in region 2 have the form

Ez = C cos kbz +D sin kbz (5.17a)

Hz = E sin kaz + F cos kaz, (5.17b)

and apply the appropriate boundary conditions, the solution for the reflection coefficients

for horizontal (A) and vertical (B) polarized electric fields can be determined. The exact

120

horizontal and vertical reflection coefficients at the surface of the grounded uniaxial dielectric

layer of thickness t with dielectric permittivity tensor given in (5.1) are

Rh = A =kz0 sin kat+ jka cos katkz0 sin kat− jka cos kat

ej2kz0t (5.18a)

Rv = B =εtkz0 cos kbt− jkb sin kbtεtkz0 cos kbt+ jkb sin kbt

ej2kz0t (5.18b)

where kz0=k0 cos θ0, ka=k0√εt − sin2 θ0, and kb=k0

√εt − εt/εz sin2 θ0. The explicit ex-

pressions for the reflection coefficients in (4.65) and (5.18) have not been found in the

literature.

A simple check of (5.2), (5.3), and (5.5) reveals that when the filling fraction approaches

zero (0), the permittivities εt and εz approach the permittivity of a homogeneous back-

ground, εa. When the filling fraction approaches one (1), the permittivities εt and εz

approach the permittivity of the materials blocks or inclusions, εb. Consequently, the so-

lutions for the horizontal and vertical reflection coefficients in (5.18) reduce to the exact

horizontal and vertical reflection coefficients at the surface of the grounded dielectric layer

of thickness t and relative permittivity εr found in (4.65a) and (4.65b).

As the filling fraction varies from one (1) to zero (0) for electrically small cells, the phase

angle of the horizontal and vertical reflection coefficients determined using the method of

moments solution should vary between the exact value obtained from a grounded dielectric

layer and the exact value obtained from a ground plane alone. The phase angles of the

horizontal and vertical reflection coefficients for grounded substrates having filling fractions

between these bounds are calculated in columns two through five of Table 5.2 using the

IE/MoM solution employing polarization currents. The values listed in columns two and

three are calculated using 192 unknown subsectional bases (Nx=Ny=Nz=4 for each of the

three polarizations) and in columns four and five using 288 unknown subsectional bases

(Nx=Ny=4 and Nz=6 for each polarization). As was mentioned at the end of Chapter 4,

fewer unknowns per wavelength are required to model the transverse variation to obtain

a given accuracy because the fields are more smoothly varying in the transverse direction.

The phase angles for the equivalent grounded uniaxial medium are calculated using the

analytical solutions in (5.18) and are listed in columns six and seven. The longitudinal

and transverse components of the permittivity tensor calculated from (5.2) and (5.6) are

listed in columns eight and nine of Table 5.2. Both solutions are obtained for a grounded

dielectric layer of relative permittivity εr=2.56 with thickness t=0.15λd, incident angles

φ0=45 and θ0=45, and unit cell size of a=0.25λ0 at z0=20t. Again, the solutions were

121

Table 5.2: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of rel-ative permittivity εr=2.56 and dielectric thickness t=0.15λd as a function offilling fraction b/a for 192 unknowns and 288 unknowns for Np=Nq=61, φ0=45,θ0=45, and a=0.25λ0

IE/MoM IE/MoM Uniaxial

(Nx=Ny=Nz=4) (Nx=Ny=4, Nz=6)

b/a Rh Rv Rh Rv Rh Rv εz εt

1.00 -67.00 87.30 -66.20 88.67 -64.99 89.52 2.56 2.56

0.90 -63.42 95.15 -62.87 95.20 -61.43 95.25 2.26 2.10

0.75 -59.90 103.77 -59.60 103.30 -58.40 102.05 1.88 1.65

0.50 -56.56 115.48 -56.48 114.79 -55.95 112.16 1.39 1.25

0.25 -55.01 123.33 -55.00 123.83 -54.90 121.27 1.10 1.06

0.00 -54.59 125.41 -54.59 125.41 -54.59 125.41 1.00 1.00

determined using at most 3721 Floquet modes (Np=Nq=61). However, the error criterion

given in (4.76) of εc < 0.001 was often satisfied by far fewer Floquet contributions. For the

data shown in Tables 5.2–5.4, the magnitudes of Rh and Rv are close to the expected value

of 1. Although the IE/MoM solution procedure is valid for any number of subsectional

bases, the effort to evaluate the impedance matrix elements for unit cell spacings larger

than a=0.25λ0 can become so large as to be computationally difficult. Fortunately, the

accuracy of the emulated uniaxial model increases with decreasing unit cell size (electrically

small unit cells). The accuracy of the magnitude and phase of the horizontal and vertical

reflection coefficients determined from the IE/MoM solutions suggests the model for the

equivalent uniaxial medium is valid.

In Table 5.3, the computation of the horizontal and vertical reflection coefficients is

carried out for the same parameter set as in the previous table but for an asymmetric

incident angle (φ0=60 and θ0=30) and is seen to again compare favorably with the solution

obtained for the uniaxial model.

In Table 5.4, the computation of the horizontal and vertical reflection coefficients is car-

ried out for the same parameters as Table 5.2 but for a dielectric constant of εr=6.15 and

is seen to again compare favorably with the solution obtained for the uniaxial model. How-

ever, the solution obtained using only 192 unknowns (Nx=Ny=Nz=4 for each of the three

polarizations) is less accurate than the case tabulated in Table 5.2. This is to be expected

122

Table 5.3: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of relativepermittivity εr=2.56 and dielectric thickness t=0.15λd as a function of fillingfraction b/a for 188 unknowns, Np=Nq=61, φ0=60, θ0=30, and a=0.25λ0

IE/MoM Uniaxial

b/a Rh Rv Rh Rv εz εt

1.00 76.75 -113.02 78.49 -111.78 2.56 2.56

0.90 80.58 -107.70 82.70 -106.71 2.26 2.10

0.75 84.62 -101.67 86.30 -101.56 1.88 1.65

0.50 88.52 -94.52 89.24 -95.48 1.39 1.25

0.25 90.36 -90.32 90.50 -91.00 1.10 1.06

0.00 90.87 -89.13 90.87 -89.13 1.00 1.00

since the electrical length in the dielectric material increases with increasing permittivity.

The solution obtained using 288 unknowns (Nx=Ny=4 and Nz=6 for each polarization) is

more accurate and suggests the model for the equivalent uniaxial medium is valid.

Table 5.4: Phase angle for Rh and Rv at z0=20t for a grounded dielectric layer of rel-ative permittivity εr=6.15 and dielectric thickness t=0.15λd as a function offilling fraction b/a for 192 unknowns and 288 unknowns for Np=Nq=61, φ0=45,θ0=45, and a=0.25λ0

IE/MoM IE/MoM Uniaxial

(Nx=Ny=Nz=4) (Nx=Ny=4, Nz=6)

b/a Rh Rv Rh Rv Rh Rv εz εt

1.00 -86.95 63.27 -88.05 64.12 -86.45 65.42 6.15 6.15

0.90 -85.45 71.96 -84.17 72.52 -80.74 74.19 5.17 3.80

0.75 -81.19 81.21 -80.50 80.84 -78.05 79.83 3.90 2.37

0.50 -77.51 93.40 -77.42 92.43 -76.57 86.96 2.29 1.46

0.25 -76.15 101.71 -76.15 101.27 -76.05 96.76 1.32 1.11

0.00 -75.91 104.09 -75.91 1104.09 -75.91 104.09 1.00 1.00

123

5.4 Rectangular Patch Antenna on a Positive Uniaxial Sub-

strate

Some general observations concerning patch antenna performance are in order before

the effect of anisotropy is addressed. The bandwidth and efficiency of a patch are increased

by increasing the thickness of the substrate or decreasing the relative dielectric constant.

This statement is physically reconciled by viewing the patch antenna as a resonant cavity

with two radiating slots at the end of the patch. As the radiating cavity is more loosely

bound either by increasing the substrate thickness or by decreasing the permittivity, the Q

of the resonator lessens; thus, the bandwidth widens. However, as was mentioned in the in-

troduction, integrating antenna structures on high permittivity substrates is advantageous

for circuit integration and for minimizing circuit/antenna size. The resulting consequence

of using the higher permittivity substrates is that the potential radiating energy is confined

even more tightly, narrowing the bandwidth. Higher permittivity substrates also necessi-

tates additionally thinning the substrate to reduce the formation of unwanted surface waves,

further reducing the bandwidth and efficiency.

Additionally, many of the practical substrates in use today have some amount of (uni-

axial) anisotropy. Anisotropy occurs naturally in some materials, whereas in others, it is

artificially produced in the manufacturing process. The characterization of the effect of

anisotropy on resonant length (frequency), bandwidth, and other design parameters is of

importance to the microwave community. The primary effect of uniaxial anisotropy on

patch antennas is the change in resonant length (frequency). This is significant because of

the narrow bandwidth of the patch itself. The relatively large shift in resonant frequency

produced in many of the modern substrates may actually force a rectangular patch designed

to operate at a specific frequency to radiate outside of the antenna bandwidth [74].

In Figure 5.5 is depicted an example design where the patch length L is 2.29 cm, the

patch width W is 1.9 cm, and the substrate thickness d is 0.159 cm. A coaxial probe is used

to feed the patch. It is necessary to match the probe impedance to the input impedance of

the patch. The input impedance ZA at the edge of a radiating patch is approximated [110]

as

ZA = 90ε2r

εr − 1

(L

W

)2

(Ω) (5.19)

where εr is the permittivity of the homogeneous substrate. For the example patch of

Figure 5.5, ZA=163 Ω. In order to properly match the coax probe to the patch, the probe

124

L

W

xy

z

d

(x ,y )p p

Figure 5.5: Geometry for a rectangular patch on a grounded dielectric substrate

is inset a distance ∆xp from the patch edge. The input resistance of (5.19) is reduced by

the factor cos2 (π∆xp/L) as the distance ∆xp from the edge is increased [37]. To correctly

match a 50 Ω coaxial probe to the example patch, the inset distance should be 0.40 cm

from the radiating edge.

For a rectangular patch on a homogeneous substrate with relative dielectric constant εr,

the lowest resonant frequency can be accurately predicted from [7] as

fr =c

2 (L+ 2∆L)√εeff

(5.20a)

where

∆L = 0.412d(εeff + 0.3)

(εeff − 0.256)(W/d+ 0.264)(W/d+ 0.8)

(5.20b)

and

εeff =εr + 1

2+εr − 1

21√

1 + 12d/W, for W/h >> 1. (5.20c)

Thus, for the patch design of Figure 5.5, the effective dielectric constant is εeff=2.13 and

the resonant frequency fr is predicted to be 4.194 GHz. A simulation of the patch using

Ansoft HFSS finds the lowest resonant frequency to be 4.125 GHz. This is close to the

value of 4.123 GHz calculated in [14] using a spectral domain MoM approach and also to

the measured value of 4.014 GHz given in [7]. The bandwidth BW of the resonant patch

can be approximated from [110] as

BW = 3.77εr − 1ε2r

W

L

d

λ. (5.21)

For the example design shown in Figure 5.5, the bandwidth is 1.85%.

125

When the anisotropic ratio is decreased to 0.5 (εt=2.32, εz=4.64), the resonant fre-

quency found from the finite element simulation shifts to the lower frequency of 3.015 GHz.

This value is consistent with the value of 3.032 GHz given in [14] for the same AR shift.

However, for the same AR=0.5 but εz=2.32, the resonant frequency shifts to 4.200 GHz

and the fractional change is on the order of the bandwidth of the antenna (∆f/fr=1.82).

This value is also consistent with with the value of 4.175 GHz given in [14] for the same

configuration. However, if the AR remains fixed and εz is increased to 4.64, the fractional

change in resonant frequency is significantly larger than the bandwidth. The dependence

of the resonant frequency fr and bandwidth ∆f/fr on the anisotropy for the rectangular

patch design is presented in Table 5.5.

Table 5.5: Dependence of resonant frequency on substrate permittivity

εt εz AR fr (GHz) ∆f/fr

2.32 2.32 1 4.125 0

1.16 2.32 0.5 4.200 1.82

2.32 4.64 0.5 3.015 26.9

Unlike (5.20), no general expression has been found to predict patch behavior on a

uniaxial substrate. However, it is well-known that the effect of anisotropy increases with

substrate thickness. While a significant amount of energy is coupled into the z -component

of the electric field of the TM modes for electrically small substrates, very little energy is

directed into the TE modes. However, as the substrate thickness increases, more energy

is coupled into the transverse component of the electric field that comprise the TE modes.

Consequently, the dependence on the transverse permittivity is increased and the effect

of the anisotropy more pronounced [74, 123]. Hence, for electrically thin substrates, the

dependence of design parameters such as resonant length and bandwidth should be strongly

dependent only on εz.

If, for the previous example where the anisotropic ratio has been decreased to 0.5 and

εz=4.64, the relative dielectric constant in (5.20) is replaced with the corresponding εz, the

predicted resonant frequency of the patch is 3.042 GHz. This is reasonably close to the

value determined from the simulation and to the value of 3.032 GHz given in [14]. This

observation is also consistent with results presented in [14] that the fractional change in

the resonant frequency is strongly dependent on εz. By modeling the uniaxial material as

a doubly periodic dielectric layer, the understanding of this dependence can be shown even

126

more clearly.

Consider a rectangular patch integrated on a uniaxial substrate. The uniaxial substrate

can be modelled as a homogeneous dielectric layer with periodic inclusions in two directions.

Because the axial and transverse components of the permittivity tensor of the uniaxial sub-

strate are known, the homogeneous permittivity and filling fraction of the doubly periodic

substrate can be found through (5.5) and (5.2). For an electrically thin substrate, the

effect of removing material from underneath the patch is a volumetric “averaging” of the

dielectric constant. This assumption is intuitively appealing and is probably valid for thin

substrates, but may not hold as the thickness of the substrate increases (and consequently

the dependence on the transverse permittivity).

As was pointed out earlier, the equivalent axial component of the permittivity tensor of

the emulated periodic substrate is effectively the volumetric average of the two phases. For

example, sapphire which has an AR of 0.81 (εt=9.4, εz=11.6) is modelled in the HFSS ma-

terial library as a homogeneous substrate with relative dielectric constant εr=10. However,

sapphire can be modeled by a medium consisting of periodic air inclusions in a dielec-

tric background with dielectric constant of 16.0 and filling fraction equal to 0.3 as seen in

Figure 5.4(b). If one simply takes the volume averages of the two phases as the effective

permittivity εeff of the equivalent substrate, one finds that εeff=11.6 which is very close the

value of the axial component of the permittivity tensor. This effective permittivity is not

to be confused with the equivalent homogeneous permittivity that can be determined from

(5.20). Clearly, modeling a sapphire substrate as a homogeneous substrate with an “effec-

tive” permittivity would yield a very different resonant frequency than if the uniaxial nature

of the material is taken into account. In fact, the lowest resonant frequency of the patch

shown in Figure 5.5 when integrated on a substrate of permittivity εr=10 yields a predicted

resonant frequency from (5.20) of 2.173 GHz. However, if the true uniaxial nature is taken

into account, the resonant frequency should be 2.022 GHz. Since the resonant frequency

and electrical length of the patch are directly related, one can interpret the effect of the

increased axial component as an effective lengthening of the patch. This interpretation is

confirmed when one views the patterns produced by the patch antenna.

Anisotropic effects shape the radiation pattern of the patch and thus in an array config-

uration, the coupling to other elements. For the example depicted in Figure 5.5, the E - and

H -plane patterns are determined for both homogeneous (εt=2.32, εz=2.32) and anisotropic

(εt=2.32, εz=4.64) substrates. The H -plane patterns shown in Figure 5.6(b) are relatively

independent of the AR and εz. However, the E -plane patterns shown in Figure 5.6(a)

127

Directivity (dB

)

-20

-10

00

30

60

90et=2.32, ez=2.32

εt=1.16, εz=2.32

εt=2.32, εz=4.64

(a) E -plane

Directivity (dB

)

-20

-10

0

0

30

60

90

εt=2.32, εz=2.32

εt=1.16, εz=2.32

εt=2.32, εz=4.64

(b) H -plane

Figure 5.6: Directivity patterns for reference patch with AR=1 and AR=0.5

show a strong dependence on εz. This is intuitively correct if one assumes that for thin

substrates, the dependence is on εz alone. Thus, if the true axial component is ignored for

substrates with positive anisotropic ratios, the beamwidth will be wider than designed and

consequently the mutual coupling in an array environment may affect the performance of

such an antenna. The observation that the E -plane beamwidth is affected by the anisotropy

while the H -plane beamwidth is not affected by it is consistent with approximations for the

beamwidths given in [11]. The E -plane and H -plane beamwidths can be approximated by

[11]

ΘE 2 cos−1

√7.03λ2

0

4(3L2e + d2)π2

(5.22a)

ΘH 2 cos−1

√1

2 + k0W(5.22b)

where Le is the effective length of the patch. As the effective length of the patch is increased,

the E -plane beamwidth decreases. The H -plane beamwidth is independent of Le.

5.5 Conclusions

Because uniaxial substrates are often expensive to manufacture and have limited flexibil-

ity for design, uniaxial substrates can be emulated (and easily fabricated) by incorporating

128

periodic inclusions in an otherwise homogeneous substrate. The doubly periodic structure

can be easily constructed using simple milling or etching techniques from simple inexpen-

sive, homogeneous substrates. This is significant because some common uniaxial materials

such as sapphire that are expensive to grow can be “artificially” replicated easily and in-

expensively. Additionally, the artificial nature of the periodic uniaxial substrate permits

the creative design of new substrates with the expanded freedom of anisotropic ratio, back-

ground permittivity, and/or fabrication technique. The primary effect of anisotropy on

rectangular patch antennas is the change in its resonant length (frequency). This is signif-

icant because of the narrow bandwidth of the patch. The relatively large shift in resonant

frequency produced in many of the modern substrates may actually force a rectangular patch

designed to operate at a specific frequency to radiate outside of the antenna bandwidth.

In addition to providing freedom of design, modeling uniaxial materials using the doubly

periodic model provides an insightful understanding of the dependence of the performance

of the substrate on its uniaxial nature.

129

CHAPTER 6

Recommendations For Future Work

One of the more interesting physical features that can be altered in the solution of

the one- and two-dimensional periodic media is the specification of an arbitrarily shaped

dielectric. Both the IE/MoM solution and plane wave expansion method can be adapted to

provide for optimized dielectric functions. Two potential applications for effective medium

theory that promise to be of value are the extension of EMT to off-axis propagation in

two-dimensional lattices and to out-of-plane propagation in two-dimensional lattices. A

number of new applications can be developed using the solution technique implemented in

Chapter 4 and 5 including new frequency selective volumes, incorporating noncommensurate

periodicities for each layer, incorporating material implants within each layer of differing

relative permittivity (dielectric and/or metallic loading), and the extension of the solution

to large planar antenna elements. The formulation also has the flexibility of providing that

each layer have different element shape and/or permittivity.

6.1 Arbitrarily Shaped Dielectric Function

By optimizing the dielectric function, improvements in design and performance can be

achieved for applications at both optical and microwave frequencies. The optimization is

accomplished through the use of genetic algorithms, neural networks, and other “smart”

algorithms. The design of a unique dielectric function is achieved by allowing the opti-

mizing algorithm or morphing program to vary the different parameters, such as unit cell

size, filling fraction, element shape, and/or relative permittivity, until a desired response

is obtained. Using the IE/MoM solution, unique configurations can be formed by selec-

tively removing individual polarization currents from the solution, commonly referred to

in antenna engineering as thinning. This is accomplished in a similar fashion to thinning

130

the number of elements in a phased array antenna. For the plane wave expansion method,

the optimization is accomplished by computing the fast Fourier transform (FFT) of the

discretized dielectric function. Each time the dielectric function is changed, a new FFT is

performed to determine the new Fourier coefficients.

Figure 6.1: Optimized frequency selective layer (volume)

New materials with unique spectral characteristics are constantly being developed and

can be incorporated into existing designs that in turn can be optimized for performance.

An example of a structure that might be produced by an optimizing algorithm is seen in

Figure 6.1, where specific individual blocks of material have been removed from a large

unit cell. The optimized structure might have application as a frequency selective layer

(volume) for use in radome applications. Additionally, lattices of dielectric elements with

differing dielectric constants can be included to provide additional freedom in the design.

In the IE/MoM solution, the polarization currents incorporate the permittivity change.

In the plane wave expansion method, a new dielectric function must be developed. As

illustrated in Figure 6.2, electrical loading of the layer is realized by incorporating two

dielectric constant 1

dielectric constant 2

Figure 6.2: Dielectric and/or metallic loading

different dielectrics or perhaps including metallic regions in the design. Currently, an entire

131

class of PBG materials, metallic photonic band-gap (MPBG) materials composed of metallic

lattice elements, are being investigated [62].

Another application where optimizing the dielectric function might be useful is supercell

theory. A supercell is a unique geometrical configuration comprised of two or more unit

cells, each with same structure. An example of a supercell can be seen in Figure 6.3 where

the individual unit cell of Figure 6.1 has been periodically repeated. In order that the

spectral response of the supercell layer be distinct from the response of a uniform layer of

effective permittivity, the spacing between the individual elements must be on the order of

a quarter of a wavelength. Consequently, applications that use the supercell design have

the disadvantage of being electrically and, depending on the frequency, physically large.

Although the response of the supercell might be excellent, the electrical and physical size

of the resulting structure might by impractical.

Figure 6.3: Frequency selective supercell

6.2 Extensions of Effective Medium Theory

To simplify the computation of the band structure for two-dimensionally periodic media,

effective medium theory was applied in Chapter 3 to reduce the two-dimensional periodic

structure to a one-dimensional equivalent structure. Using EMT, each periodic row is re-

placed by a thin homogeneous layer of effective permittivity [50] which is determined solely

as a function of the geometrical and electrical parameters of the lattice. Unfortunately, tra-

ditional EMT is restricted to near normal incidence precluding its use for many applications

of interest. Consider the two-dimensional lattice with dielectric constant ε2 immersed in a

background with dielectric constant ε1 illustrated in cross-section in Figure 6.4. Assume a

plane wave is incident in a direction along a ray direction defined by an angle φ with respect

132

x

y

ε eff

ε eff

ε eff

ε eff

ε2

ε1

E(H)φ

Figure 6.4: Higher-order effective medium theory

to the x axis (φ = 0). An extension of EMT to off-axis propagation would allow EMT to

be useful in some planar antenna applications. Another effective EMT extension would be

to model the hybrid modes that are formed for off-plane propagation in two-dimensional

lattices.

6.3 Noncommensurate Periodicities

a

b

Figure 6.5: Noncommensurate periodicities

The solution of scattering from a multilayered medium is often accomplished using a

generalized scattering matrix (GSM) technique [136] which relates the modes that propagate

133

in one layer to the corresponding modes in adjacent layers. The solution of scattering from

a multilayered medium where each layer has a unique period has only recently be solved and

then only for commensurate periodicities [12]. An example of a multilayered medium where

each layer has a unique period is illustrated in Figure 6.5. The solution found in [12] relates

the local Floquet modes within each layer to the global Floquet modes in the structure.

The solution is straightforward but relies on a tremendous amount of bookkeepping. A new

solution to this problem would be of tremendous value.

6.4 Extensive Parametric Study of Antenna Performance Near

a Periodic Structure

An extensive parametric study of the factors that impact the performance of an antenna

integrated on a periodic surface would be valuable to the microwave community. The

salient properties of the antenna, such as its input impedance, bandwidth, beamwidth,

directivity, gain, efficiency, polarization, and mutual coupling in an array environment, are

all determined by the antenna’s electrical size, physical configuration, and the environment

in which it is located. For planar resonant patches alone, the designer has a multiplicity

of factors that must be considered. These include, but are not limited to, the material

properties of the substrate such as dielectric constant, loss tangent, anisotropy, and the ease

of fabrication; the physical considerations of patch shape, patch size, and system integration;

and for realizable antennas, the practical considerations of feed location, feeding technique,

and design cost.

Although some attention has been directed at patch antenna performance on uniaxial

substrates, it has been limited to idealized cases and simple observations. The need exists

for a complete treatment of antennas located near periodic surfaces and structures. This

can only be accomplished if extensive parametric studies are performed to determine the re-

sulting effects. For example, planar antennas mounted on substrates of significant thickness

(often used to increase bandwidth) radiate most of their energy into the substrate creating

surface waves that propagate in specific directions. The reduction or elimination of these

surface waves in planar antenna application has been the Holy Grail of microwave antenna

engineers for many years. If an appropriate design for a periodic substrate that supports

the planar antenna can be found that reduces or eliminates the formation of these waves,

significant improvements in performance will be achieved.

As the need for very wideband, omni-directional antennas for use in mobile communi-

134

cation networks grows at an increasing rate, so does the need to develop new fabrication

technologies, advanced materials engineering, and novel antenna architectures to meet the

challenge. Tomorrow’s RF engineer has the task of developing new and useful ideas in

addition to developing previously found materials and concepts to their fullest potential.

135

APPENDICES

136

APPENDIX A

Transmission Line Formulation for the One-Dimensional

Periodic Array of Dielectric Slabs

Transmission line theory allows one to formulate a simple solution for the propagation

of transverse electromagnetic waves (TEM) through periodic dielectric slabs. The dielectric

slabs and free-space regions are modeled by short sections of transmission line as seen in

Figure A.1(a) where b is the length of the shorter section BB′ with characteristic impedance

Zd and propagation constant γd=jβd, a is the length of the entire transmission line section

AA′, Z0 is the characteristic impedance of the free-space transmission line sections, jβ0 is

the propagation constant of the free-space sections, and d = (b− a)/2. Due to the periodic

nature of the structure, the propagation constant through the structure can be determined

by analyzing one unit cell. For waves propagating through the structure, the voltage VA′

and current IA′ at the A′ plane is required to be the same, except for a propagation factor,

as the value of VA and IA at the A plane, or

VA′ = VAe−γa and IA′ = I−γaA (A.1)

where γ = α + jβ is the propagation constant of the line and a is the length of the line.

The input impedance at the A and A′ planes is defined as

Zp =VA

IA=

VA′

IA′. (A.2)

The equivalent transmission line model is shown in Figure A.1(b).

137

Z

+

_0 Z0Zd

A A’B B’I A I A’

+

_

VA VA’

b

a

(a) Loaded transmission line circuit

Z

+

_p

A A’I A I A’

+

_

VA VA’

a

(b) Equivalent transmission line circuit

Figure A.1: Loaded and equivalent transmission line circuits

The ABCD matrix for the transmission line in Figure A.1(a) is

A = cosh γdb cos 2β0d+ j(Z2d + Z

20

2ZdZ0

)sinh γdb sin 2β0d (A.3)

B = jZ0 cosh γdb sin 2β0d+ sinh γd

(Z2d

Z0cos2 β0d− Z2

0

Zdsin2 β0d

)(A.4)

C =j

Z0cosh γdb sin 2β0d+ sinh γd

(Z2

0

Zdcos2 β0d− Zd

Z20

sin2 β0d

)(A.5)

D = cosh γdb cos 2β0d+ j(Z2d + Z

20

2ZdZ0

)sinh γdb sin 2β0d. (A.6)

The ABCD matrix for the equivalent transmission line in Figure A.1(b) is

A = cosh γa (A.7)

B = Zp sinh γa (A.8)

C =1Zp

sinh γa (A.9)

D = cosh γa (A.10)

where Zp is the effective input impedance of the equivalent section and γ is the propagation

constant of the line.

138

The propagation constant γ of the equivalent circuit, and thus, for the periodic struc-

ture, is obtained easily from the two ABCD matrices to be the solution of the following

transcendental equation

cosh γa = cosh γdb cos 2β0d+ j(Z2d + Z

20

2ZdZ0

)sinh γdb sin 2β0d (A.11)

which can be solved using a numerical solver such as MATLAB. The input impedance of

the line can be found by equating the ratio of (A.4) and (A.5) and the ratio of (A.8) and

(A.9)

Zp = Z0

jZ0 cosh γdb sin 2β0d+ sinh γdb

(Z2

dZ0

cos2 β0d− Z20

Zdsin2 β0d

)jZ0

cosh γdb sin 2β0d+ sinh γdb(Z2

0Zd

cos2 β0d− Z2d

Z0sin2 β0d

)

1/2

. (A.12)

The frequencies where the right hand side of (A.11) returns a value greater than unity

and γ is purely real (γ = α) define the stopband of the structure. At frequencies where

cosh γa < 1, the propagation constant is purely imaginary (γ = jβ). These frequencies

define the passband of the structure. The results obtained from these equations yield

solutions that agree with the results obtained using the method moments solution and the

solution of the exact eigenvalue equation. Additionally, the solutions obtained in (A.11)

and (A.12) are identical to (9-153) and (9-154) in [78] with the exception that the solutions

contained herein are more general, being derived for a short section of reactively loaded line

of length b, as opposed to an inductively loaded case as in [78].

Other useful forms, analogies, and derivations of the exact dispersion relation for one-

dimensionally periodic dielectric slabs can be found in [16, 55, 107, 26].

139

APPENDIX B

Bravais Lattices and the Brillouin Zone

An important concept in propagation through two- and three-dimensional periodic me-

dia are direct and reciprocal lattices. For two-dimensional periodic structures, the direct

lattice must belong to one of the five two-dimensional Bravais (or space) lattices illus-

trated in Figure B.3 on page 143. Three-dimensional lattices (crystals) have 32 unique

space lattices (with three dimensions of freedom) that can be formed. A discussion of

three-dimensional crystalline structures is beyond the scope of this work and the reader is

referred to an introductory text on solid state physics such as Kittel [44].

Direct lattice

The direct lattice in Figure B.1(a) is defined by the length of the two primitive lattice

vectors a and b and the angle γ between the vectors in the plane perpendicular to a × b.

The unit cell of Figure B.1 is completed by the dashed lines, has an area of |a||b| sin γ, and

γb

a

(a) Direct lattice

b*

a*

(b) Reciprocal lattice

Figure B.1: Direct lattice defined using the primitive lattice vectors a and b and the re-ciprocal lattice defined using the primitive reciprocal lattice vectors a∗ andb∗

140

is invariant under the translation ma+nb for any integers m,n. For the two-dimensional

structures implemented in this work, the primitive lattice vectors represent the physical

distance between the periodic elements (dielectric rods or air columns).

Reciprocal lattice

In many instances the solution of a problem defined on a periodic lattice is more con-

venient to implement in Fourier space. Thus, it is convenient to define a reciprocal lattice

space. The reciprocal lattice in Figure B.1(b) is defined by two primitive reciprocal lattice

vectors a∗ and b∗. The construction of a∗ and b∗ is simple — the only requirement being

that a ·a∗=b ·b∗=2π and a ·b∗=a∗ ·b=0. As a consequence of using plane waves to expand

the periodic function, the normalization is set equal to 2π.

Example 1. Square lattice of Figure B.3(b) with spacing d

a = −d y b = d x

a∗ = −2πd

y b∗ =2πd

x

Example 2. Hexagonal lattice of Figure B.3(c) with spacing d

a = −d2

x− d√3

2y b = d x

a∗ = − 4πd√3

y b∗ =2πd

x− 2πd√3

y

Clearly, for both examples, a · a∗=b · b∗=2π and a · b∗=a∗ · b=0.

Brillouin Zone

Periodic structures, particularly those with two- and three-dimensional periodicity, are

by definition highly symmetric. Consequently, propagation through such media necessar-

ily contains many redundant propagation vectors. The unique vectors can be found and

categorized systematically using the concept of the Brillouin zone (BZ) [44] shown for a

square regular lattice in Figure B.2. For the square lattice, all of the possible directions of

propagation are grouped into eight regions with specific symmetry.2 However, symmetry

considerations reduce the number of unique regions of propagation to one. The colored re-

gion in the figure is called the irreducible Brillouin zone and represents the smallest region

where propagation within the lattice is unique. This region is further defined by symmetric2For the hexagonal lattice of Example 2, the Brillouin zone contains six groupings.

141

ΓX

M

k=k x

k=π/a x+k y

k=k x+k y

x

x y

y

0 π/a

π/a

0

Figure B.2: Irreducible Brillouin Zone (BZ)

points within the lattice denoted Γ, M, and X. The Γ coordinate is defined by (kx, ky)=(0, 0),

the M coordinate by (kx, ky)=(2π/a, 0), and the X coordinate by (kx, ky)=(2π/a, 2π/a). In

order to determine whether a mode is allowed to propagate, every possible vector that is

located within the irreducible Brillouin zone must be checked. Fortunately, the band struc-

ture can be determined approximately by sampling the edges of the zone along the Γ–M line

for modes of the form k = kxx+0 y, the M–X line for modes of the form k = 2π/a x+ kyy

and the Γ–X line for modes of the form k = kxx+ kyy.

142

ab

γ

(a) Oblique lattice: |a| = |b|; γ = 90

abγ

(b) Square lattice: |a| = |b|; γ = 90

ab

γ

(c) Hexagonal lattice: |a| = |b|; γ = 120

abγ

(d) Rectangular lattice: |a| = |b|;γ = 90

b

a

γ

(e) Centered rectangular lattice: |a| = |b|; γ = 90

Figure B.3: Two-dimensional Bravais (or space) lattices

143

APPENDIX C

Impedance Matrix Elements for Two-Dimensional Periodic

Structures

TMz Case

Piecewise constant expansion / Piecewise constant testing

Zmn =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.1)

δZ = − δmn∆x∆y

jk0Y0 (εr − 1). (C.2)

Piecewise linear expansion / Piecewise linear testing

Z11 =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.3)

Z1n =jk0Z0∆2

x∆2y

ac

∑p,q

sinc4(kxp∆x

2

)sinc4

(kyq∆y

2

)e−jkxp(x1−xn)e−jkyq (y1−yn)

k20 − k2

xp− k2

yq

(C.4)

Z1N =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)e−jkxp(x1−xN )e−jkyq (y1−yN )

k20 − k2

xp− k2

yq

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 (C.5)

144

Zm1 =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)e−jkxp(xm−x1)e−jkyq (ym−y1)

k20 − k2

xp− k2

yq

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 (C.6)

Zmn =jk0Z0∆2

x∆2y

ac

∑p,q

sinc4(kxp∆x

2

)sinc4

(kyq∆y

2

)e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.7)

ZmN =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)e−jkxp(xm−xN )e−jkyq (ym−yN )

k20 − k2

xp− k2

yq

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 (C.8)

ZN1 =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)e−jkxp(xN−x1)e−jkyq (yN−y1)

k20 − k2

xp− k2

yq

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 (C.9)

ZNn =jk0Z0∆2

x∆2y

ac

∑p,q

sinc4(kxp∆x

2

)sinc4

(kyq∆y

2

)e−jkxp(xN−xn)e−jkyq (yN−yn)

k20 − k2

xp− k2

yq

(C.10)

ZNN =jk0Z0∆2

x∆2y

ac

∑p,q

sinc2(kxp∆x

2

)sinc2

(kyq∆y

2

)1

k20 − k2

xp− k2

yq

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 (C.11)

δZ = − δmn∆x∆y

jk0Y0 (εr − 1). (C.12)

TEz Case

Piecewise linear expansion / Piecewise linear testing

Zxx11 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)1

k20 − k2

xp− k2

yq

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 (C.13)

145

Zxx1n =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(x1−xn)e−jkyq (y1−yn)

k20 − k2

xp− k2

yq

(C.14)

Zxx1N =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(x1−xN )e−jkyq (y1−yN )

k20 − k2

xp− k2

yq

(C.15)

Zxxm1 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xm−x1)e−jkyq (ym−y1)

k20 − k2

xp− k2

yq

(C.16)

Zxxmn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

timese−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.17)

ZxxmN =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(xm−xN )e−jkyq (ym−yN )

k20 − k2

xp− k2

yq

(C.18)

ZxxN1 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xN−x1)e−jkyq (yN−y1)

k20 − k2

xp− k2

yq

(C.19)

ZxxNn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xN−xn)e−jkyq (yN−yn)

k20 − k2

xp− k2

yq

(C.20)

146

ZxxNN =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.21)

Zxy11 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.22)

Zxy1n =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(x1−xn)e−jkyq (y1−yn)

k20 − k2

xp− k2

yq

(C.23)

Zxy1N =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(x1−xN )e−jkyq (y1−yN )

k20 − k2

xp− k2

yq

(C.24)

Zxym1 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xm−x1)e−jkyq (ym−y1)

k20 − k2

xp− k2

yq

(C.25)

Zxymn =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.26)

ZxymN =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(xm−xN )e−jkyq (ym−yN )

k20 − k2

xp− k2

yq

(C.27)

147

ZxyN1 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xN−x1)e−jkyq (yN−y1)

k20 − k2

xp− k2

yq

(C.28)

ZxyNn =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xN−xn)e−jkyq (yN−yn)

k20 − k2

xp− k2

yq

(C.29)

ZxyNN =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.30)

Zyx11 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.31)

Zyx1n =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(x1−xn)e−jkyq (y1−yn)

k20 − k2

xp− k2

yq

(C.32)

Zyx1N =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(x1−xN )e−jkyq (y1−yN )

k20 − k2

xp− k2

yq

(C.33)

Zyxm1 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xm−x1)e−jkyq (ym−y1)

k20 − k2

xp− k2

yq

(C.34)

148

Zyxmn =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.35)

ZyxmN =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(xm−xN )e−jkyq (ym−yN )

k20 − k2

xp− k2

yq

(C.36)

ZyxN1 =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xN−x1)e−jkyq (yN−y1)

k20 − k2

xp− k2

yq

(C.37)

ZyxNn =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xN−xn)e−jkyq (yN−yn)

k20 − k2

xp− k2

yq

(C.38)

ZyxNN =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.39)

Zyy11 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.40)

Zyy1n =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(x1−xn)e−jkyq (y1−yn)

k20 − k2

xp− k2

yq

(C.41)

149

Zyy1N =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(x1−xN )e−jkyq (y1−yN )

k20 − k2

xp− k2

yq

(C.42)

Zyym1 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xm−x1)e−jkyq (ym−y1)

k20 − k2

xp− k2

yq

(C.43)

Zyymn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.44)

ZyymN =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 e−jkxp(xm−xN )e−jkyq (ym−yN )

k20 − k2

xp− k2

yq

(C.45)

ZyyN1 =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1 + jkxp∆x − ejkxp∆x

)(kxp∆x

)2(1 + jkyq∆y − ejkyq∆y

)(kyq∆y

)2 e−jkxp(xN−x1)e−jkyq (yN−y1)

k20 − k2

xp− k2

yq

(C.46)

ZyyNn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc4

(kxp∆x

2

)sinc4

(kyq∆y

2

)

× e−jkxp(xN−xn)e−jkyq (yN−yn)

k20 − k2

xp− k2

yq

(C.47)

ZyyNN =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

×(1− jkxp∆x − e−jkxp∆x

)(kxp∆x

)2(1− jkyq∆y − e−jkyq∆y

)(kyq∆y

)2 1k20 − k2

xp− k2

yq

(C.48)

δZxx = − δmn∆x∆y

jk0Y0 (εr − 1). (C.49)

δZyy = − δmn∆x∆y

jk0Y0 (εr − 1). (C.50)

150

Piecewise constant expansion / Piecewise constant testing

Zxxmn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

xp

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.51)

Zxymn =jk0Z0∆2

x∆2y

ac

∑p,q

(−kxpkyq

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.52)

Zyxmn = Zxymn (C.53)

Zyymn =jk0Z0∆2

x∆2y

ac

∑p,q

(1− k2

yq

k20

)sinc2

(kxp∆x

2

)sinc2

(kyq∆y

2

)

× e−jkxp(xm−xn)e−jkyq (ym−yn)

k20 − k2

xp− k2

yq

(C.54)

δZxx = − δmn∆x∆y

jk0Y0 (εr − 1). (C.55)

δZyy = − δmn∆x∆y

jk0Y0 (εr − 1). (C.56)

151

APPENDIX D

Parallel-Plate Mode Reduction In Conductor-Backed Slots

Using Periodic Dielectric Substrates

Periodic dielectric substrates offer the possibility of changing the propagation charac-

teristics of planar circuits and antennas. A number of applications for such materials can

be imagined and have been implemented including various slow-wave structures, dielectric

mirrors, resonant cavities, and frequency selective surfaces (FSS). A number of researchers

have recently begun to design electromagnetic crystal structures for use in planar antenna

applications, particularly for use as reflectors in planar dipole antenna structures. One of the

early demonstrations of the potential for these materials in the microwave and millimeter-

wave band was shown by Brown et al. in [18, 17] while investigating the radiation prop-

erties of a planar dipole on a photonic crystal substrate. Cheng et al. [21] followed by

optimizing planar dipole antennas by using PBG crystals as a perfectly reflecting planar

substrate. Kesler et al. [41] designed finite thickness slabs of two-dimensional PBG ma-

terials for use as an effective reflection plane for planar dipole antennas. The reflection

and transmission properties of two- and three-dimensional PBG materials were studied by

Sigalas et al. [95, 97] for calculating the radiation properties of planar dipole antennas.

Leung et al. [53] measured the radiation patterns of a slot antenna placed on a layer-by-

layer photonic band gap crystal. For a slot operating at a frequency in the band-gap of the

three-dimensional PBG crystal, energy which would have been radiated into the substrate

is reflected. However, at the interface between the PBG and the air, the periodicity of the

PBG is broken and a parasitic mode (surface state) can exist. These surface states decrease

the efficiency by stripping power away from the radiating element.

By fabricating a resonant slot over a reflecting back plate and filling the resulting

parallel-plate with an appropriately designed periodic dielectric substrate, noticeable en-

152

hancements in both radiation pattern and bandwidth are achieved using a significantly

lower profile than traditional designs. Measured and simulated data for conductor-backed

slots with homogeneous substrates and with periodic dielectric substrates are compared.

In order to reduce backside radiation and increase the gain of planar slot antennas,

traditional designs typically place some type of reflecting surface or cavity behind the slot.

Unfortunately, increasing the profile of the slot negates one advantage of the planar radiating

element. In addition, placing reflecting surfaces behind the slot reduces the efficiency by

creating parasitic modes and using a cavity-backed design often necessitates narrowing the

bandwidth. If the slot antenna is backed by a metal plate to increase the front-to-back

ratio, parallel-plate waveguide (PPW) modes will be excited, both decreasing the efficiency

and distorting the pattern. To completely block radiation from the backside of the slot from

propagating to the finite edges of the resulting parallel-plate cavity, the cavity can be filled

with a two-dimensional periodic dielectric substrate.

The theory and design of the periodic dielectric structure placed behind the conductor-

backed slot is developed in Section D.1. In Chapter 3, the theory of electromagnetic plane

wave propagation through simple two-dimensional periodic dielectric structures is developed

both analytically (Fourier series solution) and numerically (IE/MoM solution). The plane

wave expansion solution expands the propagating electric field as a periodic function with

a prescribed phase shift and expands the periodic dielectric rods as another periodic func-

tion with no phase shift. Consequently, the electric field and dielectric rods are expanded

in Fourier series and inserted into the wave equation. The resulting band structures can

easily be determined by solving the the resulting matrix equation for the eigenvalues of the

system. In the MoM model, the dielectric rods are replaced by equivalent (polarization)

volume currents. The total field is determined as the sum of a known incident TEM wave

propagating in a direction transverse to the cylinder axis with its electric field parallel to

the cylinder axis and a scattered field which is due to radiation by the equivalent currents

induced in the dielectric by the incident field. A nontrivial solution for the field requires

the determinant of the impedance matrix to be zero, which results in a characteristic equa-

tion. The eigenvalues (propagation constants) are obtained from the roots of this equation.

The folded slot antenna design and fabrication are outlined in Section D.2. Observations

concerning the performance of the periodic dielectric structure and its use for backing slot

antennas is detailed in Section D.3. General observations about the usefulness and possible

applications for these special materials are found in the conclusions.

153

D.1 Electromagnetic Band-Gap Theory and Design

If the reflecting plate shown in Figure D.1 is located near enough to the slot such that the

reflecting plate

electromagneticband-gap (EBG) substrate

coaxially-fed folded slot

Figure D.1: Conductor-backed folded slot with periodic dielectric substrate

operating frequency is below the cutoff of the first TE/TM mode, only the dominant TEM

mode (TM0) with zero cutoff frequency will propagate. A single TEM mode is produced

by keeping the separation distance between the plates less than half the guide wavelength.

Because the propagation characteristics of the TEM mode of the resulting waveguide are

very similar to the propagation characteristics of a uniform plane wave propagating normal

to the axis of a infinite height dielectric structure, the substrate can be modeled as a

doubly-periodic array of infinite dielectric rods in free space. The rods are imaged infinitely

in height by using a PEC (perfect electric conductor) boundary and infinitely in width by

using an PMC (perfect magnetic conductor) boundary located one-half unit cell away from

the center of the rods (Figure D.2).

D.1.1 Substrate Design

For a simple two-dimensional periodic array of dielectric cylinders, the parameter com-

binations of unit cell size, element size, element shape (square/circle), lattice class (square,

triangular, hexagonal), and dielectric contrast between the insert and the background ma-

terial are the significant contributors to the resulting band structure.

154

input port

output port

PMC

PMC

output port

PEC

PEC

Figure D.2: Simulation unit cell

Lattice shape

A square lattice of dielectric columns is chosen to both simplify the fabrication and

because it produces a larger gap for relatively lower frequencies (for parallel (TM) polar-

ization) than do traditional hexagonal or honeycomb lattices [116, 63]. A brief discussion

of other two-dimensional lattice structures can be found in Appendix B. For dielectric rod

spacings of fractions of a wavelength, the primary TM band-gap of interest is the lowest

band. The lower the lowest frequency band, the more applicable the periodic dielectric

structure is for compact circuit applications.

Element shape

Although early two-dimensional research focused on using circular rods, square rods of

commensurate size are chosen to simplify the formulation and fabrication. Band structures

calculated in Chapter 3 justify replacing the traditional circular rods by square rods in

this application. The geometries of the circular and square dielectric elements are shown

in Figure D.3. Using circular rods of diameter b=4.8 mm and square rods of edge length

b=4.8 mm produces a similar band structure (first gap) for the combination of unit cell size

a=1.2 cm and relative dielectric constant εr = 10.2. In Figure D.4, one can clearly see that

both the circular and square rods have relative large first gap-to-midgap ratios of 0.176 and

0.167, respectively. Consequently, small errors in the fabrication of the substrate can be

155

a

b

(a)

a

b

(b)

Figure D.3: Unit cell dimensions for (a) square dielectric rods and (b) circular dielectricrods where for the square rod, a is the unit cell size, b is the element edgelength, and for the circular rod, a is the unit cell size and b is the diameter

Normalized insert size (x/a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7SquareCircularSubstrate design

Figure D.4: Band-gap plot as a function normalized insert size b/a

absorbed with little or no consequence. The final design of the periodic structure is shown

as a circle in the center of the gaps and produces an omni-directional stopband between 7.5

and 10 GHz. The full band structure of a square lattice of dielectric columns as a function

of b/a is seen in Figure D.5 for circular or square rods.

Design validation for finite lattice

In parallel to the the method of moments solution and plane wave (Fourier series –

eigenvalue) expansion solution, finite element simulations have been implemented to cor-

156

0.00 3.14 6.28 9.42

Nor

mal

ized

fre

quen

cy (

fa/c

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

SquareCircular

Γ βx (βy=0) X βy (βx=π/a) M βx=βy Γ

Normalized propagation constant ( βa)

Figure D.5: Full band structure of a square lattice of dielectric columns as a function ofnormalized insert size b/a

roborate the band structure of a realizable (finite) structure. In order to achieve a useful

stopband, a large (ideally, infinite) number of periods are required. However, significant

attenuations can still be achieved using only a small number of periods by designing an ap-

propriate periodic structure. The periodically dielectric-loaded parallel-plate is simulated

using the Agilent EEsof EDA High-Frequency Structure Simulator (HFSS). For a square

lattice of five periods, the simulations take between four seconds for the Γ–X direction of

the irreducible Brillouin zone1 (Figure D.6) and eight seconds for the Γ–M direction per

frequency point on a 400 MHz Pentium II PC. Because the speed of the simulation is so

Γ(k =0,k =0)

X(k =π/a,k =0)

M(k =π/a,k =π/a)x

x

xy

y

y

Figure D.6: Irreducible Brillouin Zone (BZ)

1The concept of the irreducible Brillouin zone for two-dimensional periodic structures is developed furtherin Appendix B.

157

rapid, these structures can be repetitively designed and validated quickly.

In the finite structure, the center frequency of the first stopband for the square rods

is 8.5 GHz (Γ–X direction, b/a=0.4, εr=10.2). As expected, the center frequency of the

first stopband for the circular rods is 9.0 GHz (Γ–X direction, b/a=0.4, εr=10.2), about 5%

higher than the case using square elements. However, the 10 dB bandwidth of the stopband

is 4.0 GHz for square dielectric elements and 4.7 GHz for circular dielectric elements. The

significant increase in the gap-to-midgap ratio (bandwidth) for the finite structure can be

attributed to the certain widening of the stopband – a necessary consequence of using a finite

number of elements – and to the requirement of a finite attenuation inside the stopband. At

the center frequency of the band-gap, omni-directional attenuations of at least 20 dB and

upwards of 45 dB attenuation in specific directions can be obtained using only 3–4 periods

(Figure D.7). This is compared to the ideally infinite attenuations within the stopband of

an unrealizable infinite structure.

Transmission (dB)

-20-15-10-5056789101112

Transmission (dB)

-20 -15 -10 -5 0

G MX

Fre

quen

cy (

GH

z)

1234

1234

56789101112

Fre

quen

cy (

GH

z)

Figure D.7: Simulated (HFSS) transmission spectra for Γ–X (left) and Γ–M (right) direc-tions and calculated (MoM) band diagram (center) of a two-dimensional EBGwith a=1.2 cm, b=4.8 mm, and εr=10.2 for TM polarization.

D.2 Slot Antenna Design and Fabrication

A conductor-backed folded slot with a homogeneous substrate was designed initially to

provide an acceptable match when center-fed. Preliminary work using this λg/2 reference

slot revealed the difficulty in matching the high input impedance of the slot to the coaxial

feed. Consequently, the single λg/2 slot was replaced with a folded λg slot. Not only is the

folded λg slot easier to match to the input impedance of the coaxial feed than the λg/2 slot

158

but it also has increased bandwidth.

D.2.1 Reference Slot

A half-wave reference folded slot of length 14.3 mm and width of 2.0 mm shown in

Figure D.8, where slot dimensions in parentheses refer particularly to the reference slot

offset feed

18.0 mm(14.3 mm)

2.0 mm

center feed 2.3 mm

4.0 mm

0.8 mm

4.8 mm

12.0 mm

Figure D.8: Top view of conductor-backed folded slot design with periodic dielectric sub-strate (a=1.2 cm, b=4.8 mm, and εr=10.2) where slot dimension in parenthesesrefers only to the reference slot design on an homogeneous substrate

design, was fabricated using wet etching on a square copper clad RT/duroid homogeneous

substrate with dielectric constant of 2.2, a 127 mm (5 in) edge length, and a thickness of

1.65 mm (65 mil). The reference slot was then center-fed using a simple coaxial line [46].

Note that the reference slot is simply a truncated conductor-backed slot with a homogeneous

dielectric substrate. The periodically-machined dielectric shown in the figure corresponds

to the following folded slot design.

D.2.2 Folded Slot

Similarly to the reference slot, a folded slot of length 18.0 mm and width of 2.0 mm was

fabricated using wet etching on a square copper clad RT/duroid substrate with dielectric

constant of 2.2, a 127 mm (5 in) edge length, but a thickness of only 127 µm (5 mil). A top

view of the conductor-backed folded slot design integrated with periodic dielectric substrate

159

(a=1.2 cm, b=4.8 mm, and εr=10.2) is shown in Figure D.8. The difference in size of the

two slots allows for resonances at equivalent frequencies. In order to design the slot antenna

for optimal performance, the slot is designed to resonate near the center frequency of the

designed substrate band-gap. When the folded slot design is integrated with the substrate,

as shown in Figure D.8, the energy from the slot is coupled into the parallel-plate mode

which is then stored in the dielectric lattice, loading the slot inductively. Consequently, an

off-center feed is used to provide a better impedance match.

The thin substrate was then bonded to a square copper clad RT/duroid substrate with

dielectric constant of 10.2, a 127 mm (5 in) edge length, and a thickness of 2.54 mm (100 mil)

milled as shown in Figure D.8. A small amount of dielectric (≈ 10 mil) was left on the lower

plate in order to provide some stability for mounting the slot plate. Simulations have shown

that the remaining small amounts of dielectric (≈ 10% of the total substrate height) on the

back plate change the center frequency and attenuation per period of the band-gap slightly

(≈ 5%). This observation is similar to the small change in the propagation constant of a

partially-loaded parallel-plate waveguide. For instance, the low frequency approximation

for the propagation constant β in an partially-loaded parallel plate waveguide is [24]

β =

√εrb

a+ εr(b− a)k0 =√εek0 (D.1)

where a is the dielectric sheet thickness, b is the plate separation, and εe is the effective

dielectric constant. For a frequency of 10 GHz, plate separation of 2.54 mm, dielectric sheet

thickness of 0.254 mm, and dielectric constant 10.2, the effective dielectric constant εe is

estimated to be 1.1, corresponding to less than a 5% change the propagation constant.

To provide additional support to mount the slot plate, material at the outer edge of

the substrate was not milled as can also be seen in Figure D.8. HFSS simulations have

shown that this supporting material does not significantly affect the performance of the

slot (see Section D.3). The frequency response of the fabricated folded slot design, with

noticeable resonances at 9.4 GHz and 9.6 GHz, is displayed in Figure D.9. The slots were

designed to radiate at 9.2 GHz. Unfortunately, difficulty in bonding the slot plate to the

perforated substrate may have introduced additional unwanted modes and/or a shifting of

the slot resonance. In the same figure, the finite-element simulation (HFSS) of the folded

slot design finds a resonance slightly below 9.0 GHz. The difference between the simulations

and the experiment is due to the simplifications introduced in the structure analyzed by

HFSS to reduce computer memory and simulation time and the fabrication errors mentioned

previously.

160

Frequency (GHz)

5 6 7 8 9 10 11

|S11

| (dB

)

-25

-20

-15

-10

-5

0

MeasuredSimulated

Figure D.9: Measured and simulated frequency response of conductor-backed slot with pe-riodic dielectric substrate

D.3 Results and Discussion

Simulations of coaxially-fed slots in a metal plate in free space have relatively large

10 dB bandwidths on the order of 10–15%. However, since the slot radiates equally into

each half-space, the front-to-back ratio is 0 dB. If a metal sheet is used as a reflector, the

front-to-back ratio is increased, but unwanted energy is trapped in parallel-plate modes

and radiates away from the slot. Simulations of a finite-sized reflector-backed slot with an

absorbing boundary condition at the edges of the resulting cavity, effectively modeling an

infinite parallel-plate, yield a 10 dB bandwidth of 30%. The large bandwidth is the result of

the slot radiating most of its power into the infinite waveguide. If, however, the infinite sheet

is replaced by a finite one, an undulating field pattern will be seen. This is particularly true

for ground planes that are larger than one-half wavelength [46]. Consequently, the 127 mm

edge length of the sheet in this design, corresponding to about 4 free space wavelengths,

yields significant pattern degradation. Notice the classic interference pattern (7–8 dB “dips”

in the pattern) reported earlier [13, 79] to be caused by the radiation from the edges of the

finite ground can be clearly seen in the pattern for the reference in Figure D.10. Some

suppression technique must be implemented to eliminate this parasitic radiation.

In order to reduce the effects of the finite ground, a number of suppression techniques

are known, including integrating the antenna on a substrate lens [31]. Although useful for

161

-20

-10

0-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

E plane (co-pol)E plane (x-pol)H plane (co-pol)H plane (x-pol)

Figure D.10: Measured normalized E - and H -plane antenna patterns (co- and x-pol) ofreference slot at 9.7 GHz

providing unidirectional radiation patterns by suppressing substrate surface wave formation,

lenses are not necessarily low-profile. Cavity-backed slots are another effective method of

increasing the front-to-back ratio. A simulation of a λ/4 cavity-backed slot [46] yielded a

10 dB bandwidth of 7.5%. Careful design can minimize the necessarily narrower bandwidth

of cavity-backed slots. If vertical integration space is a premium, cavity-backed designs

much like a lens may not be appropriate. Recently, a slot on a synthesized three-dimensional

metallic photonic band-gap crystal was fabricated by composing alternating layers of thin

metallic rods [53]. Unfortunately the response of the slot is very sensitive to the placement

of the slot over the rods. Gains of 2–3 dB were reported with low cross-polarization levels

for specific slot locations and for narrow frequency bands. The bandwidth and frequency

response of the antenna was not reported. Although the three-dimensional electromagnetic

crystal acts as a good reflector, it does not enhance the transmission signal as much as

was expected. As was mentioned earlier, parasitic surface states, which reduce the radiated

power, can exist when the periodicity of the three-dimensional structure is broken.

The field distribution inside a slot-fed finite parallel-plate is similar to that of a slot-fed

metallic cavity with PMC edge boundaries. Reflections from the edges of the parallel-plate

waveguide effectively create an over-moded cavity. This parasitic radiation from the edges

is dependent on the mode that is formed in the cavity. Energy leakage from the substrate

162

through the edges of the cavity manifests itself in unwanted effects such as reduced front-

to-back ratio, reduced efficiency, increased cross-polarization level, and pattern distortion

as shown in the reference antenna pattern of Figure D.10. This effect is significantly re-

duced when the specially designed periodic dielectric substrate is implemented. If the slot

is designed to radiate at a frequency inside the band-gap, the parallel-plate TEM mode

will be trapped in the dielectric lattice. Consequently, very little power is lost to substrate

modes. An effective cavity is formed, since the energy in the parallel-plate mode is re-

flected back to the radiating slot (Figure D.11). The field structure inside the parallel plate

waveguide shown in Figure D.11 was determined from a HFSS simulation implemented

by Chappell [91]. The darker regions in the figure are areas of higher field intensity; the

lighter regions are areas of lower field intensity. As mentioned previously, the supporting

material remaining at the edge of the substrate does not affect the performance of the slot.

The removed material was replaced with a metallic boundary condition and the simulations

were repeated. No significant change in field strength or field structure was observed. The

periodic dielectric substrate has two benefits over traditional metallic cavities. Namely, the

reflection from the cavity boundaries can be controlled – adding more layers of periodic

material around the slot will increase the reflectivity of the boundaries – and the boundary

conditions of the resulting effective cavity are frequency dependent.

Figure D.11: Evanescent field structure inside EBG-backed substrate (left) and propagatingmode in homogeneously-filled parallel-plate (right)

As can be clearly seen in Figure D.12, the slot has a front-to-back ratio of more than

15 dB. Low cross-polarization levels were measured in both the E and H planes. The 10 dB

bandwidth is measured to be 7.5%, lower than that of the single planar slot and similar

to the bandwidth of traditional cavity-backed designs as expected. It is observed that the

antenna patterns are similar to that of a slot in an infinite ground plane; the significant

163

-20

-10

0-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

E plane (co-pol)E plane (x-pol)H plane (co-pol)H plane (x-pol)

Figure D.12: Measured normalized E - and H -plane antenna patterns (co- and x-pol) ofconductor-backed slot with EBG substrate at 9.7 GHz

7–8 dB “dips” in the reference slot pattern are reduced to 1–2 dB in the pattern of the slot

backed by the periodic dielectric (Figure D.13).

-20

-10

0-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

E plane (EMC)E plane (reference)

Figure D.13: Measured normalized E -plane antenna patterns (co-pol) of the reference slotand the slot backed with the periodic dielectric at 9.7 GHz

164

D.3.1 Gain Measurement

A common method for gain measurement is the relative gain measurement technique

or gain-transfer (gain-comparison) method [8]. This approach is based on the substitution

of a test antenna (unknown gain) with an antenna with a known gain (standard gain) and

comparing the corresponding received power levels. Since all of the parameters are fixed

except for the gain of the receiver antenna, it can be easily shown that the Friis Transmission

equation reduces to

Gu =P urP srGs (D.2)

where Gu and Gs are the gains of the unknown and standard antennas, respectively, and

P ru and P rs are the received powers for the antenna under test and the standard gain an-

tenna, respectively. The maximum received powers (E -plane) for the slotted antenna and

for a Narda standard gain horn antenna were measured and the gain of the conductor-

backed antenna mounted over the artificial dielectric substrate antenna was found to be

approximately 3.1±0.2 dB.

The field patterns of the slot antenna mounted on the EBG-backed substrate shown in

the preceding figures are similar to those obtained for an ideal slot. For an ideal, perfectly

matched, one-half wavelength slot mounted in an perfectly conducting infinite plate, the

variation of Eθ as a function of θ can be approximated by

Eθ =cos [(π/2) cos θ]

sin θ(D.3)

and the variation of Eθ as a function of φ is constant.

D.3.2 Directivity Measurement

Directivity is a indication of the directional properties of an antenna. The directivity

can be found by finding the maximum increase in power density in a given direction from

a fixed transmitting antenna, relative to the power density with the same transmit power

distributed equally in all directions [109]. Accurate determinations of the directivity require

that the full three-dimensional antenna pattern be known or measured. Because a full (θ, φ)

pattern cannot be obtained easily for all antennas, a number of different methods have been

proposed to approximate the directivity. The predicted directivity (gain) for an ideal slot

in infinite ground is 1.6 (2.2 dB) [46]. For the ideal slot radiating only into the upper

hemisphere, the directivity (gain) is doubled to approximately 3.2 (5.2 dB) [13].

165

Principal-plane or half-power beamwidth approximation

An approximate value for the directivity D of antenna can be found by dividing the

directivity-beamwidth product DB by the principal-plane beamwidths, HPE and HPH ,

D =DB

HPEHPH. (D.4)

Appropriate values for DB range from 4π(180/π)2 = 41 253 deg2 for a rectangular beam

with no-sidelobe pattern to more realistic values of 30 000 [8] and 26 000 [110, 109] for

practical antenna measurements. Using the latter value of 26 000 as a worst-case scenario

for the directivity-beamwidth product and determining the principal-plane beamwidths of

the folded slot to be 160 and 60, a coarse estimate for the directivity is 2.7 (4.3 dB)

yielding an efficiency of about 80%.

x

y

z

φ

θ

E-plane

H-plane

Figure D.14: Axis and pattern cut definitions of the measurement systems

Pattern integration

Although half-power beamwidth measurements are simple antenna measurements that

provide a coarse estimate for the directivity, more accurate calculations can be performed

by integrating known θ and φ cuts. Assuming the radiation intensity of a given antenna

is separable in θ and φ, an estimate for the total radiated power can be determined by

integrating Eθ and Eφ over the E -plane (θ) shown in Figure D.14 while assuming the φ-

variation is constant. By integrating the pattern over θ and multiplying the result by 2π,

an estimate for the directivity is 2.76 (4.4 dB) yielding an efficiency of around 80%, similar

166

to the efficiency obtained previously. More accurate values of the radiated power density,

and consequently efficiency, can be obtained by introducing additional θ and φ cuts.

In order to verify that the estimated efficiencies calculated for the antenna are reason-

able, HFSS simulations of the finite-size, conductor-backed folded-slot antenna mounted

over the designed artificial dielectric were conducted by Chappell [91] that yield an effi-

ciency of over 90%, which is significantly higher than the estimates determined from the

measurements. However, it should be noted that the simulation assumed no conductor loss

which is one of the dominant loss mechanisms in the system.

D.4 Conclusions

Although relatively thick substrates were used in this chapter to demonstrate the useful-

ness of the conductor-backed folded slot with periodic dielectric substrates, similar results

can be obtained using significantly thinner substrates. This is very desirable for applications

involving packaging, such as layered circuits or vertical integration of components, particu-

larly where ground plane proximity is a concern. In addition, the effective cavity resulting

from the lattice is frequency dependent and tunable. Above and below the band-gap, the

periodic structure allows energy to propagate through virtually unimpeded. In the center of

the gap, the structure is virtually impassible to electromagnetic propagation. Near the band

edges, proper design could further increase the bandwidth of this antenna by decreasing the

attenuation per period or by allowing specific modes to propagate unattenuated.

167

APPENDIX E

Review Of The Method Of Moments

The use of the methods of moments [34] to numerically solve various electromagnetic

problems has been used effectively since the mid-1960’s. It has since become one of the

standard techniques used by computational electromagnetic (CEM) theorists because of its

accuracy and ease of implementation.

Consider the following nonhomogeneous equation

Lu = f (E.1)

where L is a linear operator, f is known vector quantity , and u is unknown vector quantity

to be determined. In electromagnetics, L is often a linear intergro-differential operator, f is

the known excitation, and u is the unknown field or current. We begin by expanding u in

a linear combination of basis (or expansion) functions

u =∑n

anun (E.2)

where an are the coefficients of the basis un. Finite computational resources require the

sum in (E.2) to be finite. Thus, the method of moments, although exact in theory, is an

approximate technique in practice. Substituting (E.2) into (E.1) one obtains

∑n

anLun = f . (E.3)

To determine the coefficients of the unknown basis, (E.3) is tested by performing an

inner product of the linear combination of basis functions with N weighting (or testing)

functions

∑n

an 〈wm, Lun〉 = 〈wm, f〉 m = 1, 2, . . . , N (E.4)

168

where the inner product 〈a,b〉 can be defined as

〈a,b〉 =∫

a · b∗ dx (E.5)

where x is a multidimensional variable and ∗ denotes the complex conjugate. (E.4) can be

written in matrix form as

[Lmn

] [an

]=[fm

], (E.6)

where

[Lmn

]=

〈w1, Lu1〉 〈w1, Lu2〉 · · · 〈w1, LuN 〉〈w2, Lu1〉 〈w2, Lu2〉 · · · 〈w2, LuN 〉

......

. . ....

〈wN , Lu1〉 〈wN , Lu2〉 · · · 〈wN , LuN 〉

, (E.7a)

[an

]=

a1

a2

...

aN

, (E.7b)

and

[fm

]=

〈w1, f〉〈w2, f〉

...

〈wN , f〉

. (E.7c)

In electromagnetics, [Lmn] is often represented as an impedance matrix [Zmn] relating the

excitation to the unknown.

If [Lmn]−1 exists, the solution for the coefficients an can be found from

[an

]=[Lmn

]−1 [fm

], (E.8)

using a number of well-known linear solution techniques.

169

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